Primes which contain only one odd digit
- Task
Show on this page those primes under 1,000 which when expressed in decimal contains only one odd digit.
- Stretch goal
Show on this page only the count of those primes under 1,000,000 which when expressed in decimal contains only one odd digit.
ALGOL 68
<lang algol68>BEGIN # find primes whose decimal representation contains only one odd digit #
INT max prime = 1 000 000;
# sieve the primes to max prime #
[ 1 : max prime ]BOOL prime;
prime[ 1 ] := FALSE; prime[ 2 ] := TRUE;
FOR i FROM 3 BY 2 TO UPB prime DO prime[ i ] := TRUE OD;
FOR i FROM 4 BY 2 TO UPB prime DO prime[ i ] := FALSE OD;
FOR i FROM 3 BY 2 TO ENTIER sqrt( UPB prime ) DO
IF prime[ i ] THEN FOR s FROM i * i BY i + i TO UPB prime DO prime[ s ] := FALSE OD FI
OD;
# show a count of primes #
PROC show total = ( INT count, INT limit, STRING name )VOID:
print( ( newline, "Found ", whole( count, 0 ), " ", name, " primes upto ", whole( limit, 0 ), newline ) );
# find the appropriate primes #
# 2 is the only even prime, so the final digit must be odd for primes > 2 #
# so we only need to check that the digits other than the last are all even #
INT show max = 1 000;
INT p1odd count := 0;
FOR n FROM 3 TO UPB prime DO
IF prime[ n ] THEN
BOOL p1odd := TRUE;
INT v := n OVER 10;
WHILE v > 0 AND ( p1odd := NOT ODD v ) DO
v OVERAB 10
OD;
IF p1odd THEN
# the prime has only 1 odd digit #
p1odd count +:= 1;
IF n <= show max THEN
print( ( whole( n, -5 ) ) );
IF p1odd count MOD 12 = 0 THEN print( ( newline ) ) FI
FI
FI
FI;
IF n = show max THEN
print( ( newline ) );
show total( p1odd count, show max, "single-odd-digit" )
FI
OD;
show total( p1odd count, UPB prime, "single-odd-digit" )
END</lang>
- Output:
3 5 7 23 29 41 43 47 61 67 83 89 223 227 229 241 263 269 281 283 401 409 421 443 449 461 463 467 487 601 607 641 643 647 661 683 809 821 823 827 829 863 881 883 887 Found 45 single-odd-digit primes upto 1000 Found 2560 single-odd-digit primes upto 1000000
F#
<lang fsharp> // Primes which contain only one odd number. Nigel Galloway: July 28th., 2021 let rec fN g=function 2->false |n when g=0->n=1 |n->fN (g/10) (n+g%2) primes32()|>Seq.takeWhile((>)1000)|>Seq.filter(fun g->fN g 0)|>Seq.iter(printf "%d "); printfn "" </lang>
- Output:
3 5 7 23 29 41 43 47 61 67 83 89 223 227 229 241 263 269 281 283 401 409 421 443 449 461 463 467 487 601 607 641 643 647 661 683 809 821 823 827 829 863 881 883 887
Go
<lang go>package main
import (
"fmt" "rcu"
)
func allButOneEven(prime int) bool {
digits := rcu.Digits(prime, 10)
digits = digits[:len(digits)-1]
allEven := true
for _, d := range digits {
if d&1 == 1 {
allEven = false
break
}
}
return allEven
}
func main() {
const (
LIMIT = 999
LIMIT2 = 999999
MAX_DIGITS = 3
)
primes := rcu.Primes(LIMIT)
var results []int
for _, prime := range primes[1:] {
if allButOneEven(prime) {
results = append(results, prime)
}
}
fmt.Println("Primes under", rcu.Commatize(LIMIT+1), "which contain only one odd digit:")
for i, p := range results {
fmt.Printf("%*s ", MAX_DIGITS, rcu.Commatize(p))
if (i+1)%9 == 0 {
fmt.Println()
}
}
fmt.Println("\nFound", len(results), "such primes.")
primes = rcu.Primes(LIMIT2)
count := 0
for _, prime := range primes[1:] {
if allButOneEven(prime) {
count = count + 1
}
}
cs := rcu.Commatize(count)
ls := rcu.Commatize(LIMIT2 + 1)
fmt.Println("\nThere are", cs, "primes under", ls, "which contain only one odd digit.")
}</lang>
- Output:
Primes under 1,000 which contain only one odd digit: 3 5 7 23 29 41 43 47 61 67 83 89 223 227 229 241 263 269 281 283 401 409 421 443 449 461 463 467 487 601 607 641 643 647 661 683 809 821 823 827 829 863 881 883 887 Found 45 such primes. There are 2,560 primes under 1,000,000 which contain only one odd digit.
Julia
If only one digit of a prime is odd, then that odd digit is the ones place digit. We don't actually need to check for an odd first digit once we exclude 2. <lang julia>using Primes
function isoneoddprime(n, base = 10)
d = digits(n ÷ base, base = base) return n != 2 && all(iseven, d)
end
found = filter(isoneoddprime, primes(1000)) println("Found $(length(found)) primes with one odd digit in base 10:") foreach(p -> print(rpad(last(p), 5), first(p) % 9 == 0 ? "\n" : ""), enumerate(found))
println("\nThere are ", count(isoneoddprime, primes(1_000_000)),
" primes with only one odd digit in base 10 between 1 and 1,000,000.")
</lang>
- Output:
Found 45 primes with one odd digit in base 10: 3 5 7 23 29 41 43 47 61 67 83 89 223 227 229 241 263 269 281 283 401 409 421 443 449 461 463 467 487 601 607 641 643 647 661 683 809 821 823 827 829 863 881 883 887 There are 2560 primes with only one odd digit in base 10 between 1 and 1,000,000.
Nim
<lang Nim>import sequtils, strutils
func isPrime(n: Positive): bool =
if n == 1: return false
if n mod 3 == 0: return n == 3
var d = 5
while d * d <= n:
if n mod d == 0:
return false
inc d, 2
if n mod d == 0:
return false
inc d, 4
result = true
func hasLastDigitOdd(n: Natural): bool =
var n = n n = n div 10 while n != 0: if (n mod 10 and 1) != 0: return false n = n div 10 result = true
iterator primesOneOdd(lim: Positive): int =
var n = 1
while n <= lim:
if n.hasLastDigitOdd and n.isPrime:
yield n
inc n, 2
let list = toSeq(primesOneOdd(1000))
echo "Found $# primes with only one odd digit below 1000:".format(list.len)
for i, n in list:
stdout.write ($n).align(3), if (i + 1) mod 9 == 0: '\n' else: ' '
var count = 0 for _ in primesOneOdd(100_000_000):
inc count
echo "\nFound $# primes with only one odd digit below 1_000_000.".format(count)</lang>
- Output:
Found 45 primes with only one odd digit below 1000: 3 5 7 23 29 41 43 47 61 67 83 89 223 227 229 241 263 269 281 283 401 409 421 443 449 461 463 467 487 601 607 641 643 647 661 683 809 821 823 827 829 863 881 883 887 Found 2560 primes with only one odd digit below 1_000_000.
Perl
<lang perl>#!/usr/bin/perl
use strict; use warnings; use ntheory qw( primes );
my @singleodd = grep tr/13579// == 1, @{ primes(1e3) }; my $million = grep tr/13579// == 1, @{ primes(1e6) }; print "found " . @singleodd .
"\n\n@singleodd\n\nfound $million in 1000000\n" =~ s/.{60}\K /\n/gr;</lang>
- Output:
found 45 3 5 7 23 29 41 43 47 61 67 83 89 223 227 229 241 263 269 281 283 401 409 421 443 449 461 463 467 487 601 607 641 643 647 661 683 809 821 823 827 829 863 881 883 887 found 2560 in 1000000
Phix
Relies on the fact that '0', '1', '2', etc are just as odd/even as 0, 1, 2, etc.
with javascript_semantics function oneodddigit(integer n) return sum(apply(sprint(n),odd))=1 end function sequence res = filter(get_primes_le(1_000),oneodddigit) printf(1,"Found %d one odd digit primes < 1,000: %V\n",{length(res),shorten(res,"",5)})
- Output:
Found 45 one odd digit primes < 1,000: {3,5,7,23,29,"...",829,863,881,883,887}
stretch
Fast skip from 11 direct to 20+, from 101 direct to 200+, etc. Around forty times faster than the above would be, but less than twice as fast as it would be without such skipping.
with javascript_semantics for m=1 to iff(platform()=JS?8:9) do integer m10 = power(10,m) sequence primes = get_primes_le(m10) integer n = 2, count = 0 while n<=length(primes) do integer p = primes[n], skip = 10 while true do p = floor(p/10) if odd(p) then p = (p+1)*skip n = abs(binary_search(p,primes)) exit end if if p=0 then count += 1 n += 1 exit end if skip *=10 end while end while printf(1,"Found %,d one odd digit primes < %,d\n",{count,m10}) end for
- Output:
Found 3 one odd digit primes < 10 Found 12 one odd digit primes < 100 Found 45 one odd digit primes < 1,000 Found 171 one odd digit primes < 10,000 Found 619 one odd digit primes < 100,000 Found 2,560 one odd digit primes < 1,000,000 Found 10,774 one odd digit primes < 10,000,000 Found 46,708 one odd digit primes < 100,000,000 Found 202,635 one odd digit primes < 1,000,000,000
Raku
<lang perl6>put display ^1000 .grep: { ($_ % 2) && .is-prime && (.comb[^(*-1)].all %% 2) }
sub display ($list, :$cols = 10, :$fmt = '%6d', :$title = "{+$list} matching:\n" ) {
cache $list; $title ~ $list.batch($cols)».fmt($fmt).join: "\n"
}</lang>
- Output:
45 matching:
3 5 7 23 29 41 43 47 61 67
83 89 223 227 229 241 263 269 281 283
401 409 421 443 449 461 463 467 487 601
607 641 643 647 661 683 809 821 823 827
829 863 881 883 887
REXX
<lang rexx>/*REXX pgm finds & displays primes (base ten) that contain only one odd digit (< 1,000).*/ parse arg hi cols . /*obtain optional argument from the CL.*/ if hi== | hi=="," then hi= 1000 /*Not specified? Then use the default.*/ if cols== | cols=="," then cols= 10 /* " " " " " " */ call genP /*build array of semaphores for primes.*/ w= 10 /*width of a number in any column. */ title= ' primes N (in base ten) that only contain one odd digit, where N <' commas(hi) if cols>0 then say ' index │'center(title, 1 + cols*(w+1) ) if cols>0 then say '───────┼'center("" , 1 + cols*(w+1), '─') found= 0; idx= 1 /*initialize # of primes found; IDX. */ $= /*list of primes that contain a string.*/
do j=2 for #-1; p= @.j; L= length(p) /*find primes with leading/trailing dig*/
z= 0
do k=L by -1 for L
if verify(substr(p, k, 1), 13579, 'M')>0 then do; z= z+1; if z>1 then iterate j /* ◄■■■■■■■ the filter.*/
end
end /*k*/
found= found + 1 /*bump the number of primes found. */
if cols<=0 then iterate /*Build the list (to be shown later)? */
c= commas(@.j) /*maybe add commas to the number. */
$= $ right(c, max(w, length(c) ) ) /*add a prime ──► $ list, allow big #*/
if found//cols\==0 then iterate /*have we populated a line of output? */
say center(idx, 7)'│' substr($, 2); $= /*display what we have so far (cols). */
idx= idx + cols /*bump the index count for the output*/
end /*j*/
if $\== then say center(idx, 7)"│" substr($, 2) /*possible display residual output.*/ if cols>0 then say '───────┴'center("" , 1 + cols*(w+1), '─') say say 'Found ' commas(found) title exit 0 /*stick a fork in it, we're all done. */ /*──────────────────────────────────────────────────────────────────────────────────────*/ commas: parse arg ?; do jc=length(?)-3 to 1 by -3; ?=insert(',', ?, jc); end; return ? /*──────────────────────────────────────────────────────────────────────────────────────*/ genP: @.1=2; @.2=3; @.3=5; @.4=7; @.5=11 /*define some low primes. */
#=5; sq.#= @.# **2 /*number of primes so far; prime square*/
do j=@.#+2 by 2 to hi-1 /*find odd primes from here on. */
parse var j -1 _; if _==5 then iterate /*J divisible by 5? (right dig)*/
if j// 3==0 then iterate /*" " " 3? */
if j// 7==0 then iterate /*" " " 7? */
do k=5 while sq.k<=j /* [↓] divide by the known odd primes.*/
if j // @.k == 0 then iterate j /*Is J ÷ X? Then not prime. ___ */
end /*k*/ /* [↑] only process numbers ≤ √ J */
#= #+1; @.#= j; sq.#= j*j /*bump # of Ps; assign next P; P square*/
end /*j*/; return</lang>
- output when using the default inputs:
index │ primes N (in base ten) that only contain one odd digit, where N < 1,000 ───────┼─────────────────────────────────────────────────────────────────────────────────────────────────────────────── 1 │ 3 5 7 23 29 41 43 47 61 67 11 │ 83 89 223 227 229 241 263 269 281 283 21 │ 401 409 421 443 449 461 463 467 487 601 31 │ 607 641 643 647 661 683 809 821 823 827 41 │ 829 863 881 883 887 ───────┴─────────────────────────────────────────────────────────────────────────────────────────────────────────────── Found 45 primes N (in base ten) that only contain one odd digit, where N < 1,000
- output when using the inputs of: 1000000 -1
Found 2,560 primes N (in base ten) that only contain one odd digit, where N < 1,000,000
Ring
<lang ring> load "stdlib.ring" see "working..." + nl see "Primes which contain only one odd number:" + nl row = 0
for n = 1 to 1000
odd = 0
str = string(n)
for m = 1 to len(str)
if number(str[m])%2 = 1
odd++
ok
next
if odd = 1 and isprime(n)
see "" + n + " "
row++
if row%5 = 0
see nl
ok
ok
next
see "Found " + row + " prime numbers" + nl see "done..." + nl </lang>
- Output:
working... Primes which contain only one odd number: 3 5 7 23 29 41 43 47 61 67 83 89 223 227 229 241 263 269 281 283 401 409 421 443 449 461 463 467 487 601 607 641 643 647 661 683 809 821 823 827 829 863 881 883 887 Found 45 prime numbers done...
Sidef
<lang ruby>func primes_with_one_odd_digit(upto, base = 10) {
var list = [] var digits = @(^base)
var even_digits = digits.grep { .is_even }
var odd_digits = digits.grep { .is_odd && .is_coprime(base) }
list << digits.grep { .is_odd && .is_prime && !.is_coprime(base) }...
for k in (0 .. upto.len(base)-1) {
even_digits.variations_with_repetition(k, {|*a|
next if (a.last == 0)
break if ([1, a...].digits2num(base) > upto)
odd_digits.each {|d|
var n = [d, a...].digits2num(base)
list << n if n.is_prime
}
})
}
list.sort
}
with (1e3) {|n|
var list = primes_with_one_odd_digit(n)
say "There are #{list.len} primes under #{n.commify} which contain only one odd digit:"
list.each_slice(9, {|*a| say a.map { '%3s' % _ }.join(' ') })
}
say
for k in (1..8) {
var count = primes_with_one_odd_digit(10**k).len
say "There are #{'%6s' % count.commify} such primes <= 10^#{k}"
}</lang>
- Output:
There are 45 primes under 1,000 which contain only one odd digit: 3 5 7 23 29 41 43 47 61 67 83 89 223 227 229 241 263 269 281 283 401 409 421 443 449 461 463 467 487 601 607 641 643 647 661 683 809 821 823 827 829 863 881 883 887 There are 3 such primes <= 10^1 There are 12 such primes <= 10^2 There are 45 such primes <= 10^3 There are 171 such primes <= 10^4 There are 619 such primes <= 10^5 There are 2,560 such primes <= 10^6 There are 10,774 such primes <= 10^7 There are 46,708 such primes <= 10^8
Wren
<lang ecmascript>import "/math" for Int import "/fmt" for Fmt import "/seq" for Lst
var limit = 999 var maxDigits = 3 var primes = Int.primeSieve(limit) var results = [] for (prime in primes.skip(1)) {
if (Int.digits(prime)[0...-1].all { |d| d & 1 == 0 }) results.add(prime)
} Fmt.print("Primes under $,d which contain only one odd digit:", limit + 1) for (chunk in Lst.chunks(results, 9)) Fmt.print("$,%(maxDigits)d", chunk) System.print("\nFound %(results.count) such primes.")
limit = 999999 primes = Int.primeSieve(limit) var count = 0 for (prime in primes.skip(1)) {
if (Int.digits(prime)[0...-1].all { |d| d & 1 == 0 }) count = count + 1
} Fmt.print("\nThere are $,d primes under $,d which contain only one odd digit.", count, limit + 1)</lang>
- Output:
Primes under 1,000 which contain only one odd digit: 3 5 7 23 29 41 43 47 61 67 83 89 223 227 229 241 263 269 281 283 401 409 421 443 449 461 463 467 487 601 607 641 643 647 661 683 809 821 823 827 829 863 881 883 887 Found 45 such primes. There are 2,560 primes under 1,000,000 which contain only one odd digit.
XPL0
<lang XPL0>func IsPrime(N); \Return 'true' if N is a prime number int N, I; [if N <= 1 then return false; for I:= 2 to sqrt(N) do
if rem(N/I) = 0 then return false;
return true; ];
int Count, N, Hun, Ten, One; [Count:= 0; for Hun:= 0 to 4 do
for Ten:= 0 to 4 do
for One:= 0 to 4 do
[N:= Hun*200 + Ten*20 + One*2 + 1;
if IsPrime(N) then
[IntOut(0, N);
Count:= Count+1;
if rem(Count/10) = 0 then CrLf(0) else ChOut(0, 9\tab\);
];
];
CrLf(0); IntOut(0, Count); Text(0, " such numbers found. "); ]</lang>
- Output:
3 5 7 23 29 41 43 47 61 67 83 89 223 227 229 241 263 269 281 283 401 409 421 443 449 461 463 467 487 601 607 641 643 647 661 683 809 821 823 827 829 863 881 883 887 45 such numbers found.