Ormiston pairs
Ormiston pairs is a draft programming task. It is not yet considered ready to be promoted as a complete task, for reasons that should be found in its talk page.
An Ormiston pair is two consecutive prime numbers which are anagrams, i.e. contain the same decimal digits but in a different order.
(1913, 1931) is the first such pair.
- Task
- Find and show the first 30 Ormiston pairs.
- Find and show the count of Ormiston pairs up to one million.
- Stretch
- Find and show the count of Ormiston pairs up to ten million.
- See also
ALGOL 68
When running this with ALGOL 68G, you will need to specify a large heap size with e.g., -heap 256M on the ALGOL 68G command.
Uses the "signature" idea from the XPL0 sample and the "difference is 0 MOD 18" filter from the Wren sample.
Also shows the number of prime pairs whose difference is 0 MOD 18 but are not Ormiston pairs.
BEGIN # find some Orimiston pairs - pairs of primes where the first and next #
# prime are anagrams #
PR read "primes.incl.A68" PR # include prime utilities #
INT max prime = 10 000 000; # maximum number we will consider #
INT max digits = BEGIN # count the digits of max prime #
INT v := 1;
INT d := 1;
WHILE ( v *:= 10 ) < max prime DO d +:= 1 OD;
d
END;
[ 0 : 9 ]LONG INT dp; # table of max digit powers for signature creation #
dp[ 0 ] := 1; FOR i TO UPB dp DO dp[ i ] := max digits * dp[ i - 1 ] OD;
[]BOOL prime = PRIMESIEVE max prime;
# construct a list of the primes up to the maximum prime to consider #
[]INT prime list = EXTRACTPRIMESUPTO max prime FROMPRIMESIEVE prime;
# splits n into its digits, returning the sum of their counts, each #
# scaled by the digit power of max digits #
PROC get digits = ( INT n )LONG INT:
BEGIN
INT v := n;
LONG INT result := dp[ v MOD 10 ];
WHILE ( v OVERAB 10 ) > 0 DO
result +:= dp[ v MOD 10 ]
OD;
result
END # get digits # ;
# count the Ormiston pairs #
INT o count := 0;
INT n count := 0;
INT p10 := 100 000;
FOR i TO UPB prime list - 1 DO
INT p1 = prime list[ i ];
INT p2 = prime list[ i + 1 ];
IF ( p2 - p1 ) MOD 18 = 0 THEN
# p2 and p1 might be anagrams #
IF get digits( p1 ) /= get digits( p2 ) THEN
# not an Ormiston pair afterall #
n count +:= 1
ELSE
# p1 and p2 are an Ormiston pair #
o count +:= 1;
IF o count <= 30 THEN
print( ( " (", whole( p1, -5 ), ", ", whole( p2, -5 ), ")"
, IF o count MOD 3 = 0 THEN newline ELSE " " FI
)
)
ELIF p1 >= p10 THEN
print( ( whole( o count - 1, -9 )
, " Ormiston pairs below "
, whole( p10, 0 )
, newline
)
);
p10 *:= 10
FI
FI
FI
OD;
print( ( whole( o count, -9 ), " Ormiston pairs below ", whole( max prime, 0 ), newline ) );
print( ( whole( n count, -9 ), " non-Ormiston ""0 MOD 18"" pairs bwlow ", whole( max prime, 0 ) ) )
END- Output:
( 1913, 1931) (18379, 18397) (19013, 19031)
(25013, 25031) (34613, 34631) (35617, 35671)
(35879, 35897) (36979, 36997) (37379, 37397)
(37813, 37831) (40013, 40031) (40213, 40231)
(40639, 40693) (45613, 45631) (48091, 48109)
(49279, 49297) (51613, 51631) (55313, 55331)
(56179, 56197) (56713, 56731) (58613, 58631)
(63079, 63097) (63179, 63197) (64091, 64109)
(65479, 65497) (66413, 66431) (74779, 74797)
(75913, 75931) (76213, 76231) (76579, 76597)
40 Ormiston pairs below 100000
382 Ormiston pairs below 1000000
3722 Ormiston pairs below 10000000
53369 non-Ormiston "0 MOD 18" pairs bwlow 10000000
AppleScript
use AppleScript version "2.4" -- Mac OS X 10.10 (Yosemite) or later.
use sorter : script "Insertion sort" -- <https://rosettacode.org/wiki/Sorting_algorithms/Insertion_sort#AppleScript>
on OrmistonPairs()
script o
property primes : sieveOfSundaram(13, 10000000)
end script
set output to {"First 30 Ormiston pairs:"}
set outputLine to {}
set p1 to missing value
set digits1 to {}
set counter to 0
repeat with p2 in o's primes
set p2 to p2's contents
set p to p2
set digits2 to {}
repeat until (p = 0)
set end of digits2 to p mod 10
set p to p div 10
end repeat
tell sorter to sort(digits2, 1, -1)
if (digits2 = digits1) then
if (counter < 30) then
set end of outputLine to ("{" & p1) & (", " & p2 & "}")
if ((count outputLine) = 6) then
set end of output to join(outputLine, " ")
set outputLine to {}
end if
end if
set counter to counter + 1
end if
if ((p2 > 1000000) and (p1 < 1000000)) then ¬
set end of output to "Number of pairs < 1,000,000: " & counter
set p1 to p2
set digits1 to digits2
end repeat
set end of output to "Number of pairs < 10,000,000: " & counter
return join(output, linefeed)
end OrmistonPairs
on sieveOfSundaram(lowerLimit, upperLimit)
if (upperLimit < lowerLimit) then set {upperLimit, lowerLimit} to {lowerLimit, upperLimit}
if (upperLimit < 2) then return {}
if (lowerLimit < 2) then set lowerLimit to 2
set k to (upperLimit - 1) div 2
set shift to lowerLimit div 2 - 1
script o
property sieve : makeList(k - shift, true)
on zapMultiples(n)
set i to (n * n) div 2
if (i ≤ shift) then set i to shift + n - (shift - i) mod n
repeat with i from (i - shift) to (k - shift) by n
set my sieve's item i to false
end repeat
end zapMultiples
end script
o's zapMultiples(3)
set addends to {2, 6, 8, 12, 14, 18, 24, 26}
repeat with n from 5 to (upperLimit ^ 0.5 div 1) by 30
o's zapMultiples(n)
repeat with a in addends
o's zapMultiples(n + a)
end repeat
end repeat
repeat with i from 1 to (k - shift)
if (o's sieve's item i) then set o's sieve's item i to (i + shift) * 2 + 1
end repeat
set o's sieve to o's sieve's numbers
if (lowerLimit is 2) then set o's sieve's beginning to 2
return o's sieve
end sieveOfSundaram
on makeList(limit, filler)
if (limit < 1) then return {}
script o
property lst : {filler}
end script
set counter to 1
repeat until (counter + counter > limit)
set o's lst to o's lst & o's lst
set counter to counter + counter
end repeat
if (counter < limit) then set o's lst to o's lst & o's lst's items 1 thru (limit - counter)
return o's lst
end makeList
on join(lst, delim)
set astid to AppleScript's text item delimiters
set AppleScript's text item delimiters to delim
set txt to lst as text
set AppleScript's text item delimiters to astid
return txt
end join
OrmistonPairs()
- Output:
"First 30 Ormiston pairs:
{1913, 1931} {18379, 18397} {19013, 19031} {25013, 25031} {34613, 34631} {35617, 35671}
{35879, 35897} {36979, 36997} {37379, 37397} {37813, 37831} {40013, 40031} {40213, 40231}
{40639, 40693} {45613, 45631} {48091, 48109} {49279, 49297} {51613, 51631} {55313, 55331}
{56179, 56197} {56713, 56731} {58613, 58631} {63079, 63097} {63179, 63197} {64091, 64109}
{65479, 65497} {66413, 66431} {74779, 74797} {75913, 75931} {76213, 76231} {76579, 76597}
Number of pairs < 1,000,000: 382
Number of pairs < 10,000,000: 3722"
Arturo
nextPrime: $[n][
ensure -> and? n > 2 odd? n
range .step:2 2+n ∞ | select.first => prime?
| last
]
anagrams?: $[a b] [equal? tally to :string a tally to :string b]
ormiston?: $[n] [and? -> prime? n -> anagrams? n nextPrime n]
print "First 30 Ormiston pairs:"
range .step: 2 3 ∞ | select .first:30 => ormiston?
| map 'x -> @[x nextPrime x]
| loop [a b c d e] -> print [a b c d e]
count: range .step: 2 3 1e6 | enumerate => ormiston?
print ~"\n|count| ormiston pairs less than a million"
- Output:
First 30 Ormiston pairs: [1913 1931] [18379 18397] [19013 19031] [25013 25031] [34613 34631] [35617 35671] [35879 35897] [36979 36997] [37379 37397] [37813 37831] [40013 40031] [40213 40231] [40639 40693] [45613 45631] [48091 48109] [49279 49297] [51613 51631] [55313 55331] [56179 56197] [56713 56731] [58613 58631] [63079 63097] [63179 63197] [64091 64109] [65479 65497] [66413 66431] [74779 74797] [75913 75931] [76213 76231] [76579 76597] 382 ormiston pairs less than a million
C
#include <stdio.h>
#include <stdlib.h>
#include <stdbool.h>
#include <locale.h>
bool *sieve(int limit) {
int i, p;
limit++;
// True denotes composite, false denotes prime.
bool *c = calloc(limit, sizeof(bool)); // all false by default
c[0] = true;
c[1] = true;
for (i = 4; i < limit; i += 2) c[i] = true;
p = 3; // Start from 3.
while (true) {
int p2 = p * p;
if (p2 >= limit) break;
for (i = p2; i < limit; i += 2 * p) c[i] = true;
while (true) {
p += 2;
if (!c[p]) break;
}
}
return c;
}
typedef struct {
char digs[20];
int count;
} digits;
digits getDigits(int n) {
if (n == 0) return (digits){ {0}, 1 };
digits d;
d.count = 0;
while (n > 0) {
d.digs[d.count++] = n % 10;
n = n / 10;
}
return d; // note digits are in reverse order
}
typedef struct {
int x;
int y;
} pair;
int main() {
const int limit = 1000000000;
int i, j, k, pc = 0, count = 0, count2 = 0, p1, p2, key1, key2;
digits d;
bool *c = sieve(limit);
for (i = 0; i < limit; ++i) {
if (!c[i]) ++pc;
}
int *primes = (int *)malloc(pc * sizeof(int));
for (i = 0, j = 0; i < limit; ++i) {
if (!c[i]) primes[j++] = i;
}
pair orm30[30];
int counts[5];
j = 100000;
for (i = 0; i < pc-1; ++i) {
p1 = primes[i];
p2 = primes[i+1];
if ((p2 - p1) % 18) continue;
key1 = 1;
d = getDigits(p1);
for (k = 0; k < d.count; ++k) key1 *= primes[d.digs[k]];
key2 = 1;
d = getDigits(p2);
for (k = 0; k < d.count; ++k) key2 *= primes[d.digs[k]];
if (key1 == key2) {
if (count < 30) orm30[count] = (pair){p1, p2};
if (p1 >= j) {
counts[count2++] = count;
j *= 10;
}
++count;
}
}
counts[count2] = count;
printf("First 30 Ormiston pairs:\n");
setlocale(LC_NUMERIC, "");
for (i = 0; i < 30; ++i) {
printf("[%'6d, %'6d] ", orm30[i].x, orm30[i].y);
if (!((i+1) % 3)) printf("\n");
}
printf("\n");
j = 100000;
for (i = 0; i < 5; ++i) {
printf("%'d Ormiston pairs before %'d\n", counts[i], j);
j *= 10;
}
free(c);
free(primes);
return 0;
}
- Output:
First 30 Ormiston pairs: [ 1,913, 1,931] [18,379, 18,397] [19,013, 19,031] [25,013, 25,031] [34,613, 34,631] [35,617, 35,671] [35,879, 35,897] [36,979, 36,997] [37,379, 37,397] [37,813, 37,831] [40,013, 40,031] [40,213, 40,231] [40,639, 40,693] [45,613, 45,631] [48,091, 48,109] [49,279, 49,297] [51,613, 51,631] [55,313, 55,331] [56,179, 56,197] [56,713, 56,731] [58,613, 58,631] [63,079, 63,097] [63,179, 63,197] [64,091, 64,109] [65,479, 65,497] [66,413, 66,431] [74,779, 74,797] [75,913, 75,931] [76,213, 76,231] [76,579, 76,597] 40 Ormiston pairs before 100,000 382 Ormiston pairs before 1,000,000 3,722 Ormiston pairs before 10,000,000 34,901 Ormiston pairs before 100,000,000 326,926 Ormiston pairs before 1,000,000,000
C++
#include <array>
#include <iomanip>
#include <iostream>
#include <utility>
#include <primesieve.hpp>
class ormiston_pair_generator {
public:
ormiston_pair_generator() { prime_ = pi_.next_prime(); }
std::pair<uint64_t, uint64_t> next_pair() {
for (;;) {
uint64_t prime = prime_;
auto digits = digits_;
prime_ = pi_.next_prime();
digits_ = get_digits(prime_);
if (digits_ == digits)
return std::make_pair(prime, prime_);
}
}
private:
static std::array<int, 10> get_digits(uint64_t n) {
std::array<int, 10> result = {};
for (; n > 0; n /= 10)
++result[n % 10];
return result;
}
primesieve::iterator pi_;
uint64_t prime_;
std::array<int, 10> digits_;
};
int main() {
ormiston_pair_generator generator;
int count = 0;
std::cout << "First 30 Ormiston pairs:\n";
for (; count < 30; ++count) {
auto [p1, p2] = generator.next_pair();
std::cout << '(' << std::setw(5) << p1 << ", " << std::setw(5) << p2
<< ')' << ((count + 1) % 3 == 0 ? '\n' : ' ');
}
std::cout << '\n';
for (uint64_t limit = 1000000; limit <= 1000000000; ++count) {
auto [p1, p2] = generator.next_pair();
if (p1 > limit) {
std::cout << "Number of Ormiston pairs < " << limit << ": " << count
<< '\n';
limit *= 10;
}
}
}
- Output:
First 30 Ormiston pairs: ( 1913, 1931) (18379, 18397) (19013, 19031) (25013, 25031) (34613, 34631) (35617, 35671) (35879, 35897) (36979, 36997) (37379, 37397) (37813, 37831) (40013, 40031) (40213, 40231) (40639, 40693) (45613, 45631) (48091, 48109) (49279, 49297) (51613, 51631) (55313, 55331) (56179, 56197) (56713, 56731) (58613, 58631) (63079, 63097) (63179, 63197) (64091, 64109) (65479, 65497) (66413, 66431) (74779, 74797) (75913, 75931) (76213, 76231) (76579, 76597) Number of Ormiston pairs < 1000000: 382 Number of Ormiston pairs < 10000000: 3722 Number of Ormiston pairs < 100000000: 34901 Number of Ormiston pairs < 1000000000: 326926
Delphi
Uses iterator to do both the first 30 and the millionth and 10 millionth pair. It is an easy way to make the same code do both showing a number of pairs and picking counts out of different points in the iteration.
{------- These subroutines would normally be in libraries, but they are included here for clairty------------- }
function IsPrime(N: int64): boolean;
{Fast, optimised prime test}
var I,Stop: int64;
begin
if (N = 2) or (N=3) then Result:=true
else if (n <= 1) or ((n mod 2) = 0) or ((n mod 3) = 0) then Result:= false
else
begin
I:=5;
Stop:=Trunc(sqrt(N+0.0));
Result:=False;
while I<=Stop do
begin
if ((N mod I) = 0) or ((N mod (I + 2)) = 0) then exit;
Inc(I,6);
end;
Result:=True;
end;
end;
function GetNextPrime(Start: integer): integer;
{Get the next prime number after Start}
begin
repeat Inc(Start)
until IsPrime(Start);
Result:=Start;
end;
{-----------------------------------------------------------------------------------------------------}
type TPrimeInfo = record
Prime1,Prime2: integer;
Count: integer;
end;
type TOrmIterator = class(TObject)
Info: TPrimeInfo;
private
function IsOrmistonPair(P1, P2: integer): boolean;
protected
public
procedure Reset;
function GetNext: TPrimeInfo;
end;
function TOrmIterator.IsOrmistonPair(P1,P2: integer): boolean;
{Tests if P1 and P2 primes represent Ormiston Pairs}
{I}
var S1,S2: string;
var I,J: integer;
var SL1,SL2: TStringList;
begin
Result:=False;
SL1:=TStringList.Create;
SL2:=TStringList.Create;
try
{Copy characters in numbers into string lists}
SL1.Duplicates:=dupAccept;
SL2.Duplicates:=dupAccept;
S1:=IntToStr(P1); S2:=IntToStr(P2);
for I:=1 to Length(S1) do SL1.Add(S1[I]);
for I:=1 to Length(S2) do SL2.Add(S2[I]);
{Sort them }
SL1.Sort; SL2.Sort;
{And compare them item for item - any mismatch = not Ormiston Pair }
for I:=0 to Min(SL1.Count,SL2.Count)-1 do
if SL1[I]<>SL2[I] then exit;
Result:=True;
finally Sl1.Free; SL2.Free; end;
end;
procedure TOrmIterator.Reset;
{Restart iterator}
begin
Info.Count:=0;
Info.Prime1:=1; Info.Prime2:=3;
end;
function TOrmIterator.GetNext: TPrimeInfo;
{Iterate to next Ormiston Pair}
begin
while true do
begin
Info.Prime2:=GetNextPrime(Info.Prime1);
if IsOrmistonPair(Info.Prime1,Info.Prime2) then
begin
Inc(Info.Count);
Result:=Info;
Info.Prime1:=Info.Prime2;
break;
end
else Info.Prime1:=Info.Prime2;
end;
end;
procedure ShowOrmistonPairs(Memo: TMemo);
var I: integer;
var S: string;
var OI: TOrmIterator;
var Info: TPrimeInfo;
begin
{Create iterator}
OI:=TOrmIterator.Create;
try
OI.Reset;
{Iterate throug 1st 30 pairs}
for I:=1 to 30 do
begin
Info:=OI.GetNext;
S:=S+Format('(%6D %6D) ',[Info.Prime1,Info.Prime2]);
If (Info.Count mod 3)=0 then S:=S+CRLF;
end;
Memo.Lines.Add(S);
Memo.Lines.Add('Count='+IntToStr(Info.Count));
{iterate to millionth pair}
repeat Info:=OI.GetNext
until Info.Prime2>=1000000;
Memo.Lines.Add('1000,000 ='+IntToStr(Info.Count-1));
{iterate to 10 millionth pair}
repeat Info:=OI.GetNext
until Info.Prime2>=10000000;
Memo.Lines.Add('10,000,000 ='+IntToStr(Info.Count-1));
finally OI.Free; end;
end;
- Output:
( 1913 1931) ( 18379 18397) ( 19013 19031) ( 25013 25031) ( 34613 34631) ( 35617 35671) ( 35879 35897) ( 36979 36997) ( 37379 37397) ( 37813 37831) ( 40013 40031) ( 40213 40231) ( 40639 40693) ( 45613 45631) ( 48091 48109) ( 49279 49297) ( 51613 51631) ( 55313 55331) ( 56179 56197) ( 56713 56731) ( 58613 58631) ( 63079 63097) ( 63179 63197) ( 64091 64109) ( 65479 65497) ( 66413 66431) ( 74779 74797) ( 75913 75931) ( 76213 76231) ( 76579 76597) Count=30 1000,000 =382 10,000,000 =3722 Elapsed Time: 10.046 Sec.
F#
This task uses Extensible Prime Generator (F#)
// Ormiston pairs. Nigel Galloway: January 31st., 2023
let fG(n,g)=let i=Array.zeroCreate<int>10
let rec fG n g=if g<10 then i[g]<-n i[g] 1 else i[g%10]<-n i[g%10] 1; fG n (g/10)
fG (+) n; fG (-) g; Array.forall ((=)0) i
let oPairs n=n|>Seq.pairwise|>Seq.filter fG
primes32()|>oPairs|>Seq.take 30|>Seq.iter(printf "%A "); printfn ""
printfn $"<1 million: %d{primes32()|>Seq.takeWhile((>)1000000)|>oPairs|>Seq.length}"
printfn $"<10 million: %d{primes32()|>Seq.takeWhile((>)10000000)|>oPairs|>Seq.length}"
printfn $"<100 million: %d{primes32()|>Seq.takeWhile((>)100000000)|>oPairs|>Seq.length}"
printfn $"<1 billion: %d{primes32()|>Seq.takeWhile((>)1000000000)|>oPairs|>Seq.length}"
- Output:
(1913, 1931) (18379, 18397) (19013, 19031) (25013, 25031) (34613, 34631) (35617, 35671) (35879, 35897) (36979, 36997) (37379, 37397) (37813, 37831) (40013, 40031) (40213, 40231) (40639, 40693) (45613, 45631) (48091, 48109) (49279, 49297) (51613, 51631) (55313, 55331) (56179, 56197) (56713, 56731) (58613, 58631) (63079, 63097) (63179, 63197) (64091, 64109) (65479, 65497) (66413, 66431) (74779, 74797) (75913, 75931) (76213, 76231) (76579, 76597) <1 million: 382 <10 million: 3722 <100 million: 34901 <1 billion: 326926
Factor
USING: grouping io kernel lists lists.lazy math math.parser
math.primes.lists math.statistics prettyprint sequences ;
: ormistons ( -- list )
lprimes dup cdr lzip
[ first2 [ >dec histogram ] same? ] lfilter ;
"First 30 Ormiston pairs:" print
30 ormistons ltake list>array 5 group simple-table. nl
ormistons [ first 1e6 < ] lwhile llength pprint bl
"Ormiston pairs less than a million." print
- Output:
First 30 Ormiston pairs:
{ 1913 1931 } { 18379 18397 } { 19013 19031 } { 25013 25031 } { 34613 34631 }
{ 35617 35671 } { 35879 35897 } { 36979 36997 } { 37379 37397 } { 37813 37831 }
{ 40013 40031 } { 40213 40231 } { 40639 40693 } { 45613 45631 } { 48091 48109 }
{ 49279 49297 } { 51613 51631 } { 55313 55331 } { 56179 56197 } { 56713 56731 }
{ 58613 58631 } { 63079 63097 } { 63179 63197 } { 64091 64109 } { 65479 65497 }
{ 66413 66431 } { 74779 74797 } { 75913 75931 } { 76213 76231 } { 76579 76597 }
382 Ormiston pairs less than a million.
FreeBASIC
#include "isprime.bas"
Function GetSig(Byval N As Integer) As Integer
Dim As Integer Sig = 0
Do While N > 0
Sig += 1 Shl (N Mod 10)
N \= 10
Loop
Return Sig
End Function
Dim As Integer Cnt = 0, N0 = 0, Sig0 = 0, N = 3, Sig
Do
If isPrime(N) Then
Sig = GetSig(N)
If Sig = Sig0 Then
Cnt += 1
If Cnt <= 30 Then
Print Using "##### #####"; N0; N;
If Cnt Mod 3 = 0 Then Print Else Print " ";
End If
End If
Sig0 = Sig
N0 = N
End If
If N = 1e5 -1 Then Print !"\nOrmiston pairs up to one hundred thousand: "; Cnt
If N = 1e6 -1 Then Print "Ormiston pairs up to one million: "; Cnt
If N = 1e7 -1 Then Print "Ormiston pairs up to ten million: "; Cnt: Exit Do
N += 2
Loop
Sleep- Output:
1913 1931 18379 18397 19013 19031 25013 25031 34613 34631 35617 35671 35879 35897 36979 36997 37379 37397 37813 37831 40013 40031 40213 40231 40639 40693 45613 45631 48091 48109 49279 49297 51613 51631 55313 55331 56179 56197 56713 56731 58613 58631 63079 63097 63179 63197 64091 64109 65479 65497 66413 66431 74779 74797 75913 75931 76213 76231 76579 76597 Ormiston pairs up to one hundred thousand: 40 Ormiston pairs up to one million: 382 Ormiston pairs up to ten million: 3722
Go
package main
import (
"fmt"
"rcu"
)
func main() {
const limit = 1e9
primes := rcu.Primes(limit)
var orm30 [][2]int
j := int(1e5)
count := 0
var counts []int
for i := 0; i < len(primes)-1; i++ {
p1 := primes[i]
p2 := primes[i+1]
if (p2-p1)%18 != 0 {
continue
}
key1 := 1
for _, dig := range rcu.Digits(p1, 10) {
key1 *= primes[dig]
}
key2 := 1
for _, dig := range rcu.Digits(p2, 10) {
key2 *= primes[dig]
}
if key1 == key2 {
if count < 30 {
orm30 = append(orm30, [2]int{p1, p2})
}
if p1 >= j {
counts = append(counts, count)
j *= 10
}
count++
}
}
counts = append(counts, count)
fmt.Println("First 30 Ormiston pairs:")
for i := 0; i < 30; i++ {
fmt.Printf("%5v ", orm30[i])
if (i+1)%3 == 0 {
fmt.Println()
}
}
fmt.Println()
j = int(1e5)
for i := 0; i < len(counts); i++ {
fmt.Printf("%s Ormiston pairs before %s\n", rcu.Commatize(counts[i]), rcu.Commatize(j))
j *= 10
}
}
- Output:
First 30 Ormiston pairs: [ 1913 1931] [18379 18397] [19013 19031] [25013 25031] [34613 34631] [35617 35671] [35879 35897] [36979 36997] [37379 37397] [37813 37831] [40013 40031] [40213 40231] [40639 40693] [45613 45631] [48091 48109] [49279 49297] [51613 51631] [55313 55331] [56179 56197] [56713 56731] [58613 58631] [63079 63097] [63179 63197] [64091 64109] [65479 65497] [66413 66431] [74779 74797] [75913 75931] [76213 76231] [76579 76597] 40 Ormiston pairs before 100,000 382 Ormiston pairs before 1,000,000 3,722 Ormiston pairs before 10,000,000 34,901 Ormiston pairs before 100,000,000 326,926 Ormiston pairs before 1,000,000,000
J
For this, we would like to be able to test if a prime number is the first value in an Ormiston pair:
isorm=: -:&(/:~)&":&> 4 p: ]
We could also use a routine to organize pairs of numbers as moderate width lines of text:
fmtpairs=: {{ names <@([,',',])&":/"1 y}}
Then the task becomes:
fmtpairs (,. 4 p:]) p:30{.I. isorm i.&.(p:inv) 1e6
1913,1931 18379,18397 19013,19031 25013,25031
34613,34631 35617,35671 35879,35897 36979,36997
37379,37397 37813,37831 40013,40031 40213,40231
40639,40693 45613,45631 48091,48109 49279,49297
51613,51631 55313,55331 56179,56197 56713,56731
58613,58631 63079,63097 63179,63197 64091,64109
65479,65497 66413,66431 74779,74797 75913,75931
76213,76231 76579,76597
+/isorm i.&.(p:inv) 1e6 NB. number of Ormiston pairs less than 1e6
382
+/isorm i.&.(p:inv) 1e7 NB. number of Ormiston pairs less than 1e7
3722
jq
Preliminaries
# Input: a positive integer
# Output: an array, $a, of length .+1 such that
# $a[$i] is $i if $i is prime, and false otherwise.
def primeSieve:
# erase(i) sets .[i*j] to false for integral j > 1
def erase($i):
if .[$i] then
reduce (range(2*$i; length; $i)) as $j (.; .[$j] = false)
else .
end;
(. + 1) as $n
| (($n|sqrt) / 2) as $s
| [null, null, range(2; $n)]
| reduce (2, 1 + (2 * range(1; $s))) as $i (.; erase($i)) ;The Task
def digits: tostring | explode;
def ormiston_pairs($limit):
($limit | primeSieve | map(select(.))) as $primes
| range(0; $primes|length-1) as $i
| $primes[$i] as $p1
| $primes[$i+1] as $p2
| select( ($p2|digits|sort) == ($p1|digits|sort) )
| [$p1, $p2] ;
def task($limit):
reduce ormiston_pairs($limit) as $pair (
{count:0, orm30: [], counts: [], j: 1e5};
if .count < 30 then .orm30 += [$pair] else . end
| if $pair[0] >= .j
then .counts += [.count]
| .j *= 10
else .
end
| .count += 1 )
| .counts += [.count]
| ("First 30 Ormiston pairs:", (.orm30 | map(tostring) | _nwise(3) | join(" "))),
"",
foreach range(0; .counts|length) as $i (.j = 1e5;
.emit = "\(.counts[$i]) Ormiston pairs before \(.j)"
| .j *= 10;
select(.emit).emit) );
task(1e7) # ten million- Output:
First 30 Ormiston pairs: [1913,1931] [18379,18397] [19013,19031] [25013,25031] [34613,34631] [35617,35671] [35879,35897] [36979,36997] [37379,37397] [37813,37831] [40013,40031] [40213,40231] [40639,40693] [45613,45631] [48091,48109] [49279,49297] [51613,51631] [55313,55331] [56179,56197] [56713,56731] [58613,58631] [63079,63097] [63179,63197] [64091,64109] [65479,65497] [66413,66431] [74779,74797] [75913,75931] [76213,76231] [76579,76597] 40 Ormiston pairs before 100000 382 Ormiston pairs before 1000000 3722 Ormiston pairs before 10000000
Nim
import std/[algorithm, bitops, math, strformat, strutils]
type Sieve = object
data: seq[byte]
func `[]`(sieve: Sieve; idx: Positive): bool =
## Return value of element at index "idx".
let idx = idx shr 1
let iByte = idx shr 3
let iBit = idx and 7
result = sieve.data[iByte].testBit(iBit)
func `[]=`(sieve: var Sieve; idx: Positive; val: bool) =
## Set value of element at index "idx".
let idx = idx shr 1
let iByte = idx shr 3
let iBit = idx and 7
if val: sieve.data[iByte].setBit(iBit)
else: sieve.data[iByte].clearBit(iBit)
func newSieve(lim: Positive): Sieve =
## Create a sieve with given maximal index.
result.data = newSeq[byte]((lim + 16) shr 4)
func initPrimes(lim: Positive): seq[Natural] =
## Initialize the list of primes from 2 to "lim".
var composite = newSieve(lim)
composite[1] = true
for n in countup(3, sqrt(lim.toFloat).int, 2):
if not composite[n]:
for k in countup(n * n, lim, 2 * n):
composite[k] = true
result.add 2
for n in countup(3, lim, 2):
if not composite[n]:
result.add n
let primes = initPrimes(100_000_000)
func digits(n: Positive): seq[int] =
## Return the sorted list of digits of "n".
var n = n.Natural
while n != 0:
result.add n mod 10
n = n div 10
result.sort()
echo "First 30 Ormiston pairs:"
var count = 0
var limit = 100_000
for i in 1..(primes.len - 2):
let p1 = primes[i]
let p2 = primes[i + 1]
if p1.digits == p2.digits:
inc count
if count <= 30:
stdout.write &"({p1:5}, {p2:5})"
stdout.write if count mod 3 == 0: '\n' else: ' '
if count == 30: echo()
elif p1 >= limit:
echo &"Number of Ormiston pairs below {insertSep($limit)}: {count - 1}"
limit *= 10
if limit == 100_000_000:
break
- Output:
First 30 Ormiston pairs: ( 1913, 1931) (18379, 18397) (19013, 19031) (25013, 25031) (34613, 34631) (35617, 35671) (35879, 35897) (36979, 36997) (37379, 37397) (37813, 37831) (40013, 40031) (40213, 40231) (40639, 40693) (45613, 45631) (48091, 48109) (49279, 49297) (51613, 51631) (55313, 55331) (56179, 56197) (56713, 56731) (58613, 58631) (63079, 63097) (63179, 63197) (64091, 64109) (65479, 65497) (66413, 66431) (74779, 74797) (75913, 75931) (76213, 76231) (76579, 76597) Number of Ormiston pairs below 100_000: 40 Number of Ormiston pairs below 1_000_000: 382 Number of Ormiston pairs below 10_000_000: 3722
Pascal
Free Pascal
//update the digits by adding difference.// Using MOD 18 = 0 and convert is faster.
program Ormiston;
{$IFDEF FPC}{$MODE DELPHI} {$OPTIMIZATION ON,ALL}{$ENDIF}
{$IFDEF WINDOWS}{$APPLICATION CONSOLE}{$ENDIF}
uses
sysutils,strUtils;
//********* segmented sieve of erathostenes *********
{segmented sieve of Erathostenes using only odd numbers}
{using presieved sieve of small primes, to reduce the most time consuming}
const
smlPrimes :array [0..10] of Byte = (2,3,5,7,11,13,17,19,23,29,31);
maxPreSievePrimeNum = 8;
maxPreSievePrime = 19;//smlPrimes[maxPreSievePrimeNum];
cSieveSize = 2*16384;//<= High(Word)+1 // Level I Data Cache
type
tSievePrim = record
svdeltaPrime:word;//diff between actual and new prime
svSivOfs:word; //Offset in sieve
svSivNum:LongWord;//1 shl (1+16+32) = 5.6e14
end;
tpSievePrim = ^tSievePrim;
var
//sieved with primes 3..maxPreSievePrime.here about 255255 Byte
{$ALIGN 32}
preSieve :array[0..3*5*7*11*13*17*19-1] of Byte;//must be > cSieveSize
{$ALIGN 32}
Sieve :array[0..cSieveSize-1] of Byte;
{$ALIGN 32}
//prime = FoundPrimesOffset + 2*FoundPrimes[0..FoundPrimesCnt]
FoundPrimes : array[0..12252] of Word;
{$ALIGN 32}
sievePrimes : array[0..1077863] of tSievePrim;
FoundPrimesOffset : Uint64;
FoundPrimesCnt,
FoundPrimesIdx,
FoundPrimesTotal,
SieveNum,
SieveMaxIdx,
preSieveOffset,
LastInsertedSievePrime :NativeUInt;
procedure CopyPreSieveInSieve; forward;
procedure CollectPrimes; forward;
procedure sieveOneSieve; forward;
procedure Init0Sieve; forward;
procedure SieveOneBlock; forward;
procedure preSieveInit;
var
i,pr,j,umf : NativeInt;
Begin
fillchar(preSieve[0],SizeOf(preSieve),#1);
i := 1;
pr := 3;// starts with pr = 3
umf := 1;
repeat
IF preSieve[i] =1 then
Begin
pr := 2*i+1;
j := i;
repeat
preSieve[j] := 0;
inc(j,pr);
until j> High(preSieve);
umf := umf*pr;
end;
inc(i);
until (pr = maxPreSievePrime)OR(umf>High(preSieve)) ;
preSieveOffset := 0;
end;
function InsertSievePrimes(PrimPos:NativeInt):NativeInt;
var
j :NativeUINt;
i,pr : NativeUInt;
begin
i := 0;
//ignore first primes already sieved with
if SieveNum = 0 then
i := maxPreSievePrimeNum;
pr :=0;
j := Uint64(SieveNum)*cSieveSize*2-LastInsertedSievePrime;
with sievePrimes[PrimPos] do
Begin
pr := FoundPrimes[i]*2+1;
svdeltaPrime := pr+j;
j := pr;
end;
inc(PrimPos);
for i := i+1 to FoundPrimesCnt-1 do
Begin
IF PrimPos > High(sievePrimes) then
BREAK;
with sievePrimes[PrimPos] do
Begin
pr := FoundPrimes[i]*2+1;
svdeltaPrime := (pr-j);
j := pr;
end;
inc(PrimPos);
end;
LastInsertedSievePrime :=Uint64(SieveNum)*cSieveSize*2+pr;
result := PrimPos;
end;
procedure CalcSievePrimOfs(lmt:NativeUint);
//lmt High(sievePrimes)
var
i,pr : NativeUInt;
sq : Uint64;
begin
pr := 0;
i := 0;
repeat
with sievePrimes[i] do
Begin
pr := pr+svdeltaPrime;
IF sqr(pr) < (cSieveSize*2) then
Begin
svSivNum := 0;
svSivOfs := (pr*pr-1) DIV 2;
end
else
Begin
SieveMaxIdx := i;
pr := pr-svdeltaPrime;
BREAK;
end;
end;
inc(i);
until i > lmt;
for i := i to lmt do
begin
with sievePrimes[i] do
Begin
pr := pr+svdeltaPrime;
sq := sqr(pr);
svSivNum := sq DIV (2*cSieveSize);
svSivOfs := ( (sq - Uint64(svSivNum)*(2*cSieveSize))-1)DIV 2;
end;
end;
end;
procedure sievePrimesInit;
var
i,j,pr,PrimPos:NativeInt;
Begin
LastInsertedSievePrime := 0;
preSieveOffset := 0;
SieveNum :=0;
CopyPreSieveInSieve;
//normal sieving of first sieve
i := 1; // start with 3
repeat
while Sieve[i] = 0 do
inc(i);
pr := 2*i+1;
inc(i);
j := ((pr*pr)-1) DIV 2;
if j > High(Sieve) then
BREAK;
repeat
Sieve[j] := 0;
inc(j,pr);
until j > High(Sieve);
until false;
CollectPrimes;
PrimPos := InsertSievePrimes(0);
LastInsertedSievePrime := FoundPrimes[PrimPos]*2+1;
IF PrimPos < High(sievePrimes) then
Begin
Init0Sieve;
sieveOneBlock;
repeat
sieveOneBlock;
dec(SieveNum);
PrimPos := InsertSievePrimes(PrimPos);
inc(SieveNum);
until PrimPos > High(sievePrimes);
end;
Init0Sieve;
end;
procedure Init0Sieve;
begin
FoundPrimesTotal :=0;
preSieveOffset := 0;
SieveNum :=0;
CalcSievePrimOfs(High(sievePrimes));
end;
procedure CopyPreSieveInSieve;
var
lmt : NativeInt;
Begin
lmt := preSieveOffset+cSieveSize;
lmt := lmt-(High(preSieve)+1);
IF lmt<= 0 then
begin
Move(preSieve[preSieveOffset],Sieve[0],cSieveSize);
if lmt <> 0 then
inc(preSieveOffset,cSieveSize)
else
preSieveOffset := 0;
end
else
begin
Move(preSieve[preSieveOffset],Sieve[0],cSieveSize-lmt);
Move(preSieve[0],Sieve[cSieveSize-lmt],lmt);
preSieveOffset := lmt
end;
end;
procedure sieveOneSieve;
var
sp:tpSievePrim;
pSieve :pByte;
i,j,pr,sn,dSievNum :NativeUint;
Begin
pr := 0;
sn := sieveNum;
sp := @sievePrimes[0];
pSieve := @Sieve[0];
For i := SieveMaxIdx downto 0 do
with sp^ do
begin
pr := pr+svdeltaPrime;
IF svSivNum = sn then
Begin
j := svSivOfs;
repeat
pSieve[j] := 0;
inc(j,pr);
until j > High(Sieve);
dSievNum := j DIV cSieveSize;
svSivOfs := j-dSievNum*cSieveSize;
svSivNum := sn+dSievNum;
end;
inc(sp);
end;
i := SieveMaxIdx+1;
repeat
if i > High(SievePrimes) then
BREAK;
with sp^ do
begin
if svSivNum > sn then
Begin
SieveMaxIdx := I-1;
Break;
end;
pr := pr+svdeltaPrime;
j := svSivOfs;
repeat
Sieve[j] := 0;
inc(j,pr);
until j > High(Sieve);
dSievNum := j DIV cSieveSize;
svSivOfs := j-dSievNum*cSieveSize;
svSivNum := sn+dSievNum;
end;
inc(i);
inc(sp);
until false;
end;
procedure CollectPrimes;
//extract primes to FoundPrimes
var
pSieve : pbyte;
pFound : pWord;
i,idx : NativeUint;
Begin
FoundPrimesOffset := SieveNum*2*cSieveSize;
FoundPrimesIdx := 0;
pFound :=@FoundPrimes[0];
i := 0;
idx := 0;
IF SieveNum = 0 then
//include small primes used to pre-sieve
Begin
repeat
pFound[idx]:= (smlPrimes[idx]-1) DIV 2;
inc(idx);
until smlPrimes[idx]>maxPreSievePrime;
i := (smlPrimes[idx] -1) DIV 2;
end;
//grabbing the primes without if then -> reduces time extremly
//primes are born to let branch-prediction fail.
pSieve:= @Sieve[Low(Sieve)];
repeat
//store every value until a prime aka 1 is found
pFound[idx]:= i;
inc(idx,pSieve[i]);
inc(i);
until i>High(Sieve);
FoundPrimesCnt:= idx;
inc(FoundPrimesTotal,Idx);
end;
procedure SieveOneBlock;
begin
CopyPreSieveInSieve;
sieveOneSieve;
CollectPrimes;
inc(SieveNum);
end;
function Nextprime:Uint64;
Begin
result := FoundPrimes[FoundPrimesIdx]*2+1+FoundPrimesOffset;
if (FoundPrimesIdx=0) AND (sievenum = 1) then
inc(result);
inc(FoundPrimesIdx);
If FoundPrimesIdx>= FoundPrimesCnt then
SieveOneBlock;
end;
function PosOfPrime: Uint64;inline;
Begin
result := FoundPrimesTotal-FoundPrimesCnt+FoundPrimesIdx;
end;
function TotalCount :Uint64;inline;
begin
result := FoundPrimesTotal;
end;
function SieveSize :LongInt;inline;
Begin
result := 2*cSieveSize;
end;
function SieveStart:Uint64;inline;
Begin
result := (SieveNum-1)*2*cSieveSize;
end;
procedure InitPrime;
Begin
Init0Sieve;
SieveOneBlock;
end;
//********* segmented sieve of erathostenes *********
const
Limit= 10*1000*1000*1000;
type
tDigits10 = array[0..15] of byte;
td10_UsedDgts2 = array[0..3] of Uint32;
td10_UsedDgts3 = array[0..1] of Uint64;
tpd10_UsedDgts3 = ^td10_UsedDgts3;
procedure OutIn(cnt,p1,p2:NativeInt);
Begin
write('[',Numb2USA(IntToStr(p1)):6,'|',Numb2USA(IntToStr(p2)):6,']');
if cnt MOD 5 = 0 then
writeln;
end;
function OutByPot10(cnt,prLimit:NativeInt):NativeInt;
Begin
writeln(Numb2USA(IntToStr(cnt)):12,' Ormiston pairs before ',Numb2USA(IntToStr(prLimit)):14);
result := 10*prLimit;
end;
procedure Convert2Digits10(p:NativeUint;var outP:tDigits10);
var
r : NativeUint;
begin
// fillchar(outP,SizeOf(outP),#0);//takes longer
td10_UsedDgts3(outP)[0]:=0;td10_UsedDgts3(outP)[1]:=0;
repeat
r := p DIV 10;
inc(outP[p-10*r]);
p := r;
until r = 0;
end;
function CheckOrmiston(const d1,d2:tpd10_UsedDgts3):boolean;inline;
begin
result := (d1^[0]=d2^[0]) AND (d1^[1]=d2^[1]);
end;
var
{$align 16}
p1,p2 :tDigits10;
pr,pr1,prLimit :nativeInt;
cnt : NativeUint;
Begin
preSieveInit;
sievePrimesInit;
InitPrime;
prLimit := 100*1000;
cnt := 0;
pr1 := nextprime;
repeat
pr := nextprime;
if pr > limit then
BREAK;
if (pr-pr1) mod 18 = 0 then
begin
Convert2Digits10(pr1,p1);
Convert2Digits10(pr,p2);
if CheckOrmiston(@p1,@p2) then
begin
inc(cnt);
IF cnt <= 30 then
OutIn(cnt,pr1,pr);
end;
end;
if pr >=prLimit then
prlimit:= OutByPot10(cnt,prlimit);
pr1:= pr;
until false;
OutByPot10(cnt,prlimit);
end.
- @TIO.RUN:
//only get all primes to 1E10 Real time: 13.294 s CPU share: 99.11 %
[ 1,913| 1,931][18,379|18,397][19,013|19,031][25,013|25,031][34,613|34,631]
[35,617|35,671][35,879|35,897][36,979|36,997][37,379|37,397][37,813|37,831]
[40,013|40,031][40,213|40,231][40,639|40,693][45,613|45,631][48,091|48,109]
[49,279|49,297][51,613|51,631][55,313|55,331][56,179|56,197][56,713|56,731]
[58,613|58,631][63,079|63,097][63,179|63,197][64,091|64,109][65,479|65,497]
[66,413|66,431][74,779|74,797][75,913|75,931][76,213|76,231][76,579|76,597]
40 Ormiston pairs before 100,000
382 Ormiston pairs before 1,000,000
3,722 Ormiston pairs before 10,000,000
34,901 Ormiston pairs before 100,000,000
326,926 Ormiston pairs before 1,000,000,000
3,037,903 Ormiston pairs before 10,000,000,000
Real time: 21.114 s User time: 20.862 s Sys. time: 0.057 s CPU share: 99.07 %
@home real 0m6,873s user 0m6,864s sys 0m0,008s
Perl
use strict;
use warnings;
use feature 'say';
use ntheory <primes vecfirstidx>;
my(@O,$pairs);
my @primes = @{ primes(1,1e8) };
my @A = map { join '', sort split '', $_ } @primes;
for (1..$#primes-1) { push @O, $_ if $A[$_] eq $A[$_+1] }
say "First 30 Ormiston pairs:";
$pairs .= sprintf "(%5d,%5d) ", $primes[$_], $primes[$_+1] for @O[0..29];
say $pairs =~ s/.{42}\K/\n/gr;
for (
[1e5, 'one hundred thousand'],
[1e6, 'one million'],
[1e7, 'ten million']
) {
my($limit,$text) = @$_;
my $i = vecfirstidx { $primes[$_] >= $limit } @O;
printf "%4d Ormiston pairs before %s\n", $i, $text;
}
- Output:
First 30 Ormiston pairs: ( 1913, 1931) (18379,18397) (19013,19031) (25013,25031) (34613,34631) (35617,35671) (35879,35897) (36979,36997) (37379,37397) (37813,37831) (40013,40031) (40213,40231) (40639,40693) (45613,45631) (48091,48109) (49279,49297) (51613,51631) (55313,55331) (56179,56197) (56713,56731) (58613,58631) (63079,63097) (63179,63197) (64091,64109) (65479,65497) (66413,66431) (74779,74797) (75913,75931) (76213,76231) (76579,76597) 40 Ormiston pairs before one hundred thousand 382 Ormiston pairs before one million 3722 Ormiston pairs before ten million
Phix
with javascript_semantics atom t0 = time(), t1 = time()+1 constant limit = iff(platform()=JS?1e8:1e9), -- (keep JS<10s) primes = get_primes_le(limit) sequence orm30 = {}, counts = {} integer count = 0, nc = 1e5 for i=1 to length(primes)-1 do integer p1 = primes[i], p2 = primes[i+1] if remainder(p2-p1,18)=0 and sort(sprint(p1))=sort(sprint(p2)) then if count<30 then orm30 &= {sprintf("[%5d %5d]",{p1, p2})} end if if p1>=nc then counts &= count nc *= 10 end if count += 1 end if if time()>t1 then progress("%d/%d\r",{i,length(primes)}) t1 = time()+1 end if end for progress("") counts &= count printf(1,"First 30 Ormiston pairs:\n%s\n",join_by(orm30,1,3)) for i,c in counts do printf(1,"%,d Ormiston pairs before %,d\n", {c, power(10,i+4)}) end for ?elapsed(time()-t0)
- Output:
First 30 Ormiston pairs: [ 1913 1931] [18379 18397] [19013 19031] [25013 25031] [34613 34631] [35617 35671] [35879 35897] [36979 36997] [37379 37397] [37813 37831] [40013 40031] [40213 40231] [40639 40693] [45613 45631] [48091 48109] [49279 49297] [51613 51631] [55313 55331] [56179 56197] [56713 56731] [58613 58631] [63079 63097] [63179 63197] [64091 64109] [65479 65497] [66413 66431] [74779 74797] [75913 75931] [76213 76231] [76579 76597] 40 Ormiston pairs before 100,000 382 Ormiston pairs before 1,000,000 3,722 Ormiston pairs before 10,000,000 34,901 Ormiston pairs before 100,000,000 326,926 Ormiston pairs before 1,000,000,000 "39.8s"
slower but higher limits
-- -- demo\rosetta\Ormiston_pairs.exw -- =============================== -- -- Uses a segmented sieve, which is about half the speed of get_primes_le(), but uses far less memory -- If permited, get_primes_le(1e10) would generate a result of 455,052,511 primes, more than 32 bit -- can cope with, and use over 6GB of ram, and take about 11mins 44s, that is on this box at least, -- whereas this processes them on-the-fly, and only uses about 6MB of memory (ie 0.1% of 6GB). -- with javascript_semantics atom t0 = time() procedure ormiston_pairs(atom limit) // Generate primes using the segmented sieve of Eratosthenes. // credit: https://gist.github.com/kimwalisch/3dc39786fab8d5b34fee integer segment_size = floor(sqrt(limit)), count = 0, i = 3, s = 3 atom p1 = 2, n = 3, nc = 1e5, low = 0, t1 = time()+1 sequence isprime = repeat(true,segment_size+1), primes = {}, multiples = {}, orm30 = repeat(0,30) while low<=limit do sequence sieve = repeat(true,segment_size+1) if time()>t1 then progress("Processing %,d/%,d (%3.2f%%)\r",{low,limit,(low/limit)*100}) t1 = time()+1 end if // current segment = [low, high] atom high = min(low+segment_size,limit) // generate sieving primes using simple sieve of Eratosthenes while i*i<=min(high,segment_size) do if isprime[i+1] then for j=i*i to segment_size by i do isprime[j+1] = false end for end if i += 2 end while // initialize sieving primes for segmented sieve while s*s<=high do if isprime[s+1] then primes &= s multiples &= s*s-low end if s += 2 end while // sieve the current segment for mi,j in multiples do integer k = primes[mi]*2 while j<segment_size do sieve[j+1] = false j += k end while multiples[mi] = j - segment_size end for while n<=high do if sieve[n-low+1] then // n is a prime if remainder(n-p1,18)=0 and sort(sprint(p1))=sort(sprint(n)) then if p1>=nc then string e = elapsed_short(time()-t0) progress("%,d Ormiston pairs before %,d (%s)\n", {count, nc, e}) nc *= 10 end if count += 1 if count<=30 then orm30[count] = sprintf("[%5d %5d]",{p1, n}) if count=30 then printf(1,"First 30 Ormiston pairs:\n%s\n",join_by(orm30,1,3)) end if end if end if p1 = n end if n += 2 end while low += segment_size end while string e = elapsed_short(time()-t0) progress("%,d Ormiston pairs before %,d (%s)\n", {count, nc, e}) end procedure ormiston_pairs(iff(platform()=JS?1e8:1e9))
- Output:
With limit upped to 1e10
First 30 Ormiston pairs: [ 1913 1931] [18379 18397] [19013 19031] [25013 25031] [34613 34631] [35617 35671] [35879 35897] [36979 36997] [37379 37397] [37813 37831] [40013 40031] [40213 40231] [40639 40693] [45613 45631] [48091 48109] [49279 49297] [51613 51631] [55313 55331] [56179 56197] [56713 56731] [58613 58631] [63079 63097] [63179 63197] [64091 64109] [65479 65497] [66413 66431] [74779 74797] [75913 75931] [76213 76231] [76579 76597] 40 Ormiston pairs before 100,000 (0s) 382 Ormiston pairs before 1,000,000 (0s) 3,722 Ormiston pairs before 10,000,000 (0s) 34,901 Ormiston pairs before 100,000,000 (5s) 326,926 Ormiston pairs before 1,000,000,000 (55s) 3,037,903 Ormiston pairs before 10,000,000,000 (21:57)
Note that running this under pwa/p2js with a limit of 1e9 would get you a blank screen for 1min 25s, hence I've limited it to 1e8 (8s)
I have not the patience to see whether JavaScript would actually cope with 1e10, but it should (with a blank screen for at least half an hour).
Python
""" rosettacode.org task Ormiston_pairs """
from sympy import primerange
PRIMES1M = list(primerange(1, 1_000_000))
ASBASE10SORT = [str(sorted(list(str(i)))) for i in PRIMES1M]
ORMISTONS = [(PRIMES1M[i - 1], PRIMES1M[i]) for i in range(1, len(PRIMES1M))
if ASBASE10SORT[i - 1] == ASBASE10SORT[i]]
print('First 30 Ormiston pairs:')
for (i, o) in enumerate(ORMISTONS):
if i < 30:
print(f'({o[0] : 6} {o[1] : 6} )',
end='\n' if (i + 1) % 5 == 0 else ' ')
else:
break
print(len(ORMISTONS), 'is the count of Ormiston pairs up to one million.')
- Output:
First 30 Ormiston pairs: ( 1913 1931 ) ( 18379 18397 ) ( 19013 19031 ) ( 25013 25031 ) ( 34613 34631 ) ( 35617 35671 ) ( 35879 35897 ) ( 36979 36997 ) ( 37379 37397 ) ( 37813 37831 ) ( 40013 40031 ) ( 40213 40231 ) ( 40639 40693 ) ( 45613 45631 ) ( 48091 48109 ) ( 49279 49297 ) ( 51613 51631 ) ( 55313 55331 ) ( 56179 56197 ) ( 56713 56731 ) ( 58613 58631 ) ( 63079 63097 ) ( 63179 63197 ) ( 64091 64109 ) ( 65479 65497 ) ( 66413 66431 ) ( 74779 74797 ) ( 75913 75931 ) ( 76213 76231 ) ( 76579 76597 ) 382 is the count of Ormiston pairs up to one million.
Raku
use Lingua::EN::Numbers;
use List::Divvy;
my @primes = lazy (^∞).hyper.grep( &is-prime ).map: { $_ => .comb.sort.join };
my @Ormistons = @primes.kv.map: { ($^value.key, @primes[$^key+1].key) if $^value.value eq @primes[$^key+1].value };
say "First thirty Ormiston pairs:";
say @Ormistons[^30].batch(3)».map( { "({.[0].fmt: "%5d"}, {.[1].fmt: "%5d"})" } ).join: "\n";
say '';
say +@Ormistons.&before( *[1] > $_ ) ~ " Ormiston pairs before " ~ .Int.&cardinal for 1e5, 1e6, 1e7;
- Output:
First thirty Ormiston pairs: ( 1913, 1931) (18379, 18397) (19013, 19031) (25013, 25031) (34613, 34631) (35617, 35671) (35879, 35897) (36979, 36997) (37379, 37397) (37813, 37831) (40013, 40031) (40213, 40231) (40639, 40693) (45613, 45631) (48091, 48109) (49279, 49297) (51613, 51631) (55313, 55331) (56179, 56197) (56713, 56731) (58613, 58631) (63079, 63097) (63179, 63197) (64091, 64109) (65479, 65497) (66413, 66431) (74779, 74797) (75913, 75931) (76213, 76231) (76579, 76597) 40 Ormiston pairs before one hundred thousand 382 Ormiston pairs before one million 3722 Ormiston pairs before ten million
Ring
see "working..." + nl
see "First 30 Ormiston pairs:" + nl
limit = 1000000
Primes = []
Primes1 = []
Primes2 = []
add(Primes,2)
pr = 1
while true
pr = pr + 2
if isPrime(pr) and pr < limit
add(Primes,pr)
ok
if pr > limit
exit
ok
end
n = 0
row = 0
for n = 1 to len(Primes) - 1
Primes1 = []
Primes2 = []
str1 = string(Primes[n])
str2 = string(Primes[n+1])
for p = 1 to len(str1)
add(Primes1,substr(str1,p,1))
next
for p = 1 to len(str2)
add(Primes2,substr(str2,p,1))
next
Sort1 = sort(Primes1)
Sort2 = sort(Primes2)
flag = 1
if len(Sort1) = len(Sort2)
for p = 1 to len(Sort1)
if Sort1[p] != Sort2[p]
flag = 0
exit
ok
next
if flag = 1
row++
if row < 31
str1m = str1
str2m = str2
see "(" + str1 + ", " + str2 + ") "
if row % 3 = 0
see nl
ok
ok
ok
ok
end
see nl + "Number of pairs < 1,000,000: " + row + nl
see "done..." + nl
func isPrime num
if (num <= 1) return 0 ok
if (num % 2 = 0 and num != 2) return 0 ok
for i = 3 to floor(num / 2) -1 step 2
if (num % i = 0) return 0 ok
next
return 1- Output:
working... First 30 Ormiston pairs: ( 1913, 1931) (18379, 18397) (19013, 19031) (25013, 25031) (34613, 34631) (35617, 35671) (35879, 35897) (36979, 36997) (37379, 37397) (37813, 37831) (40013, 40031) (40213, 40231) (40639, 40693) (45613, 45631) (48091, 48109) (49279, 49297) (51613, 51631) (55313, 55331) (56179, 56197) (56713, 56731) (58613, 58631) (63079, 63097) (63179, 63197) (64091, 64109) (65479, 65497) (66413, 66431) (74779, 74797) (75913, 75931) (76213, 76231) (76579, 76597) Number of pairs < 1,000,000: 382 done...
Rust
// [dependencies]
// primal = "0.3"
fn get_digits(mut n: usize) -> [usize; 10] {
let mut digits = [0; 10];
while n > 0 {
digits[n % 10] += 1;
n /= 10;
}
digits
}
fn ormiston_pairs() -> impl std::iter::Iterator<Item = (usize, usize)> {
let mut digits = [0; 10];
let mut prime = 0;
let mut primes = primal::Primes::all();
std::iter::from_fn(move || {
for p in primes.by_ref() {
let prime0 = prime;
prime = p;
let digits0 = digits;
digits = get_digits(prime);
if digits == digits0 {
return Some((prime0, prime));
}
}
None
})
}
fn main() {
let mut count = 0;
let mut op = ormiston_pairs();
println!("First 30 Ormiston pairs:");
for (p1, p2) in op.by_ref() {
count += 1;
let c = if count % 3 == 0 { '\n' } else { ' ' };
print!("({:5}, {:5}){}", p1, p2, c);
if count == 30 {
break;
}
}
println!();
let mut limit = 1000000;
for (p1, _) in op.by_ref() {
if p1 > limit {
println!("Number of Ormiston pairs < {}: {}", limit, count);
limit *= 10;
if limit == 10000000000 {
break;
}
}
count += 1;
}
}
- Output:
First 30 Ormiston pairs: ( 1913, 1931) (18379, 18397) (19013, 19031) (25013, 25031) (34613, 34631) (35617, 35671) (35879, 35897) (36979, 36997) (37379, 37397) (37813, 37831) (40013, 40031) (40213, 40231) (40639, 40693) (45613, 45631) (48091, 48109) (49279, 49297) (51613, 51631) (55313, 55331) (56179, 56197) (56713, 56731) (58613, 58631) (63079, 63097) (63179, 63197) (64091, 64109) (65479, 65497) (66413, 66431) (74779, 74797) (75913, 75931) (76213, 76231) (76579, 76597) Number of Ormiston pairs < 1000000: 382 Number of Ormiston pairs < 10000000: 3722 Number of Ormiston pairs < 100000000: 34901 Number of Ormiston pairs < 1000000000: 326926
Wren
import "./math" for Int
import "./fmt" for Fmt
var limit = 1e9
var primes = Int.primeSieve(limit)
var orm30 = []
var j = 1e5
var count = 0
var counts = []
for (i in 0...primes.count-1) {
var p1 = primes[i]
var p2 = primes[i+1]
if ((p2 - p1) % 18 != 0) continue
var key1 = 1
for (dig in Int.digits(p1)) key1 = key1 * primes[dig]
var key2 = 1
for (dig in Int.digits(p2)) key2 = key2 * primes[dig]
if (key1 == key2) {
if (count < 30) orm30.add([p1, p2])
if (p1 >= j) {
counts.add(count)
j = j * 10
}
count = count + 1
}
}
counts.add(count)
System.print("First 30 Ormiston pairs:")
Fmt.tprint("[$,6d] ", orm30, 3)
System.print()
j = 1e5
for (i in 0...counts.count) {
Fmt.print("$,d Ormiston pairs before $,d", counts[i], j)
j = j * 10
}- Output:
First 30 Ormiston pairs: [ 1,913 1,931] [18,379 18,397] [19,013 19,031] [25,013 25,031] [34,613 34,631] [35,617 35,671] [35,879 35,897] [36,979 36,997] [37,379 37,397] [37,813 37,831] [40,013 40,031] [40,213 40,231] [40,639 40,693] [45,613 45,631] [48,091 48,109] [49,279 49,297] [51,613 51,631] [55,313 55,331] [56,179 56,197] [56,713 56,731] [58,613 58,631] [63,079 63,097] [63,179 63,197] [64,091 64,109] [65,479 65,497] [66,413 66,431] [74,779 74,797] [75,913 75,931] [76,213 76,231] [76,579 76,597] 40 Ormiston pairs before 100,000 382 Ormiston pairs before 1,000,000 3,722 Ormiston pairs before 10,000,000 34,901 Ormiston pairs before 100,000,000 326,926 Ormiston pairs before 1,000,000,000
XPL0
func IsPrime(N); \Return 'true' if N is prime
int N, I;
[if N <= 2 then return N = 2;
if (N&1) = 0 then \even >2\ return false;
for I:= 3 to sqrt(N) do
[if rem(N/I) = 0 then return false;
I:= I+1;
];
return true;
];
func GetSig(N); \Return signature of N
\A "signature" is the count of each digit in N packed into a 32-bit word
int N, Sig;
[Sig:= 0;
repeat N:= N/10;
Sig:= Sig + 1<<(rem(0)*3);
until N = 0;
return Sig;
];
int Cnt, N, N0, Sig, Sig0;
[Cnt:= 0; N0:= 0; Sig0:= 0; N:= 3;
Format(6, 0);
loop [if IsPrime(N) then
[Sig:= GetSig(N);
if Sig = Sig0 then
[Cnt:= Cnt+1;
if Cnt <= 30 then
[RlOut(0, float(N0)); RlOut(0, float(N));
if rem(Cnt/3) = 0 then CrLf(0) else Text(0, " ");
];
];
Sig0:= Sig;
N0:= N;
];
if N = 1_000_000-1 then
[Text(0, "^m^jOrmiston pairs up to one million: ");
IntOut(0, Cnt);
];
if N = 10_000_000-1 then
[Text(0, "^m^jOrmiston pairs up to ten million: ");
IntOut(0, Cnt);
quit;
];
N:= N+2;
];
]- Output:
1913 1931 18379 18397 19013 19031 25013 25031 34613 34631 35617 35671 35879 35897 36979 36997 37379 37397 37813 37831 40013 40031 40213 40231 40639 40693 45613 45631 48091 48109 49279 49297 51613 51631 55313 55331 56179 56197 56713 56731 58613 58631 63079 63097 63179 63197 64091 64109 65479 65497 66413 66431 74779 74797 75913 75931 76213 76231 76579 76597 Ormiston pairs up to one million: 382 Ormiston pairs up to ten million: 3722