# Primes which contain only one odd digit

Primes which contain only one odd digit 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.

Show on this page those primes under   1,000   which when expressed in decimal contain 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 contain only one odd digit.

## 11l

Translation of: Nim
```F is_prime(n)
I n == 2
R 1B
I n < 2 | n % 2 == 0
R 0B
L(i) (3 .. Int(sqrt(n))).step(2)
I n % i == 0
R 0B
R 1B

F hasLastDigitOdd(=n)
n I/= 10
L n != 0
I ((n % 10) [&] 1) != 0
R 0B
n I/= 10
R 1B

V l = (1.<1000).step(2).filter(n -> hasLastDigitOdd(n) & is_prime(n))
print(‘Found ’l.len‘ primes with only one odd digit below 1000:’)
L(n) l
print(‘#3’.format(n), end' I (L.index + 1) % 9 == 0 {"\n"} E ‘ ’)

V count = 0
L(n) (1.<1'000'000).step(2)
I hasLastDigitOdd(n) & is_prime(n)
count++
print("\nFound "count‘ primes with only one odd digit below 1000000.’)```
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 1000000.
```

## Action!

```INCLUDE "H6:SIEVE.ACT"

BYTE FUNC OddDigitsCount(INT x)
BYTE c,d

c=0
WHILE x#0
DO
d=x MOD 10
IF (d&1)=1 THEN
c==+1
FI
x==/10
OD
RETURN (c)

PROC Main()
DEFINE MAX="999"
BYTE ARRAY primes(MAX+1)
INT i,count=[0]

Put(125) PutE() ;clear the screen
Sieve(primes,MAX+1)
FOR i=2 TO MAX
DO
IF primes(i)=1 AND OddDigitsCount(i)=1 THEN
PrintI(i) Put(32)
count==+1
FI
OD
PrintF("%E%EThere are %I primes",count)
RETURN```
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

There are 45 primes
```

## ALGOL 68

```BEGIN  # find primes whose decimal representation contains only one odd digit #
# sieve the primes to 1 000 000 #
[]BOOL prime = PRIMESIEVE 1 000 000;
# 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```
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
```

## Arturo

```onlyOneOddDigit?: function [n][
and? -> prime? n
-> one? select digits n => odd?
]

primesWithOnlyOneOddDigit: select 1..1000 => onlyOneOddDigit?

loop split.every: 9 primesWithOnlyOneOddDigit 'x ->
print map x 's -> pad to :string s 4

nofPrimesBelow1M: enumerate 1..1000000 => onlyOneOddDigit?

print ""
print ["Found" nofPrimesBelow1M "primes with only one odd digit below 1000000."]
```
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 2560 primes with only one odd digit below 1000000.```

## AWK

```# syntax: GAWK -f PRIMES_WHICH_CONTAIN_ONLY_ONE_ODD_NUMBER.AWK
BEGIN {
start = 1
stop = 999
for (i=start; i<=stop; i++) {
if (is_prime(i)) {
if (gsub(/[13579]/,"&",i) == 1) {
rec_odd = sprintf("%s%5d%1s",rec_odd,i,++count_odd%10?"":"\n")
}
if (gsub(/[02468]/,"&",i) == 1) {
rec_even = sprintf("%s%5d%1s",rec_even,i,++count_even%10?"":"\n")
}
}
}
printf("%s\nPrimes which contain only one odd number %d-%d: %d\n\n",rec_odd,start,stop,count_odd)
printf("%s\nPrimes which contain only one even number %d-%d: %d\n\n",rec_even,start,stop,count_even)
exit(0)
}
function is_prime(x,  i) {
if (x <= 1) {
return(0)
}
for (i=2; i<=int(sqrt(x)); i++) {
if (x % i == 0) {
return(0)
}
}
return(1)
}
```
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
Primes which contain only one odd number 1-999: 45

2    23    29    41    43    47    61    67    83    89
101   103   107   109   127   149   163   167   181   211
233   239   251   257   271   277   293   307   347   349
367   383   389   419   431   433   439   457   479   491
499   503   509   521   523   541   547   563   569   587
613   617   619   631   653   659   673   677   691   701
709   727   743   761   769   787   811   839   853   857
859   877   907   929   941   947   967   983
Primes which contain only one even number 1-999: 78
```

## C#

Modifies a conventional prime sieve to cull the items with more than one odd digit.

```using System;
using System.Collections.Generic;

class Program
{
// starts as an ordinary odds-only prime sieve, but becomes
// extraordinary after the else statement...
static List<uint> sieve(uint max, bool ordinary = false)
{
uint k = ((max - 3) >> 1) + 1,
lmt = ((uint)(Math.Sqrt(max++) - 3) >> 1) + 1;
var pl = new List<uint> { };
var ic = new bool[k];
for (uint i = 0, p = 3; i < lmt; i++, p += 2) if (!ic[i])
for (uint j = (p * p - 3) >> 1; j < k; j += p) ic[j] = true;
if (ordinary)
{
for (uint i = 0, j = 3; i < k; i++, j += 2)
}
else
for (uint i = 0, j = 3, t = j; i < k; i++, t = j += 2)
if (!ic[i])
{
while ((t /= 10) > 0)
if (((t % 10) & 1) == 1) goto skip;
skip:;
}
return pl;
}

static void Main(string[] args)
{
var pl = sieve((uint)1e9);
uint c = 0, l = 10, p = 1;
Console.WriteLine("List of one-odd-digit primes < 1,000:");
for (int i = 0; pl[i] < 1000; i++)
Console.Write("{0,3}{1}", pl[i], i % 9 == 8 ? "\n" : "  ");
string fmt = "\nFound {0:n0} one-odd-digit primes < 10^{1} ({2:n0})";
foreach (var itm in pl)
if (itm < l) c++;
else Console.Write(fmt, c++, p++, l, l *= 10);
Console.Write(fmt, c++, p++, l);
}
}
```
Output:
```List of one-odd-digit primes < 1,000:
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 3 one-odd-digit primes < 10^1 (10)
Found 12 one-odd-digit primes < 10^2 (100)
Found 45 one-odd-digit primes < 10^3 (1,000)
Found 171 one-odd-digit primes < 10^4 (10,000)
Found 619 one-odd-digit primes < 10^5 (100,000)
Found 2,560 one-odd-digit primes < 10^6 (1,000,000)
Found 10,774 one-odd-digit primes < 10^7 (10,000,000)
Found 46,708 one-odd-digit primes < 10^8 (100,000,000)
Found 202,635 one-odd-digit primes < 10^9 (1,000,000,000)```

## Delphi

Works with: Delphi version 6.0

```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;

procedure GetDigits(N: integer; var IA: TIntegerDynArray);
{Get an array of the integers in a number}
var T: integer;
begin
SetLength(IA,0);
repeat
begin
T:=N mod 10;
N:=N div 10;
SetLength(IA,Length(IA)+1);
IA[High(IA)]:=T;
end
until N<1;
end;

procedure ShowOneOddDigitPrime(Memo: TMemo);
var I,Cnt1,Cnt2: integer;
var IA: TIntegerDynArray;
var S: string;

function HasOneOddDigit(N: integer): boolean;
var I,Odd: integer;
begin
Odd:=0;
GetDigits(N,IA);
for I:=0 to High(IA) do
if (IA[I] and 1)=1 then Inc(Odd);
Result:=Odd=1;
end;

begin
Cnt1:=0; Cnt2:=0;
S:='';
for I:=0 to 1000000-1 do
if IsPrime(I) then
if HasOneOddDigit(I) then
begin
Inc(Cnt1);
if I<1000 then
begin
Inc(Cnt2);
S:=S+Format('%4D',[I]);
If (Cnt2 mod 5)=0 then S:=S+CRLF;
end;
end;
end;
```
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

Count < 1,000     = 45
Count < 1,000,000 = 2560
Elapsed Time: 171.630 ms.
```

## EasyLang

```fastfunc isprim num .
i = 2
while i <= sqrt num
if num mod i = 0
return 0
.
i += 1
.
return 1
.
fastfunc nextprim prim .
repeat
prim += 1
until isprim prim = 1
.
return prim
.
func digodd n .
while n > 0
r += if n mod 2 = 1
n = n div 10
.
return r
.
p = 2
repeat
if digodd p = 1
write p & " "
.
p = nextprim p
until p >= 1000
.```
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
```

## F#

This task uses Extensible Prime Generator (F#)

```// 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 ""
```
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
```

## Factor

Works with: Factor version 0.99 2021-06-02
```USING: grouping io lists lists.lazy literals math math.primes
numspec prettyprint ;

<<
DIGIT: o 357
DIGIT: q 1379
DIGIT: e 2468
DIGIT: E 02468

NUMSPEC: one-odd-candidates o eq eEq ... ;
>>

CONSTANT: p \$[ one-odd-candidates [ prime? ] lfilter ]

"Primes with one odd digit under 1,000:" print
p [ 1000 < ] lwhile list>array 9 group simple-table.

"\nCount of such primes under 1,000,000:" print
p [ 1,000,000 < ] lwhile llength .

"\nCount of such primes under 1,000,000,000:" print
p [ 1,000,000,000 < ] lwhile llength .
```
Output:
```Primes with one odd digit under 1,000:
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

Count of such primes under 1,000,000:
2560

Count of such primes under 1,000,000,000:
202635
```

## FreeBASIC

```#include "isprime.bas"

function b(j as uinteger, n as uinteger) as uinteger
'auxiliary function for evendig below
dim as uinteger m = int(log(n-1)/log(5))
return int((n-1-5^m)/5^j)
end function

function evendig(n as uinteger) as uinteger
'produces the integers with only even digits
dim as uinteger m = int(log(n-1)/log(5)), a
a = ((2*b(m, n)) mod 8 + 2)*10^m
if n<=1 or m=0 then return a
for j as uinteger = 0 to m-1
a += ((2*b(j, n)) mod 10)*10^j
next j
return a
end function

dim as uinteger n=1, count = 0, p=1
while p<1000000
p = 1 + evendig(n)   'if it's a prime the odd digit must be the last one
if isprime(p) then
count += 1
if p<1000 then print p
end if
n+=1
wend

print "There are ";count;" such primes below one million."```

## Go

Translation of: Wren
Library: Go-rcu
```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     = 9999999999
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.\n")

primes = rcu.Primes(LIMIT2)
count := 0
pow := 10
for _, prime := range primes[1:] {
if allButOneEven(prime) {
count++
}
if prime > pow {
fmt.Printf("There are %7s such primes under %s\n", rcu.Commatize(count), rcu.Commatize(pow))
pow *= 10
}
}
fmt.Printf("There are %7s such primes under %s\n", rcu.Commatize(count), rcu.Commatize(pow))
}
```
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       3 such primes under 10
There are      12 such primes under 100
There are      45 such primes under 1,000
There are     171 such primes under 10,000
There are     619 such primes under 100,000
There are   2,560 such primes under 1,000,000
There are  10,774 such primes under 10,000,000
There are  46,708 such primes under 100,000,000
There are 202,635 such primes under 1,000,000,000
There are 904,603 such primes under 10,000,000,000
```

```import Data.List (intercalate, maximum, transpose)
import Data.List.Split (chunksOf)
import Data.Numbers.Primes (primes)
import Text.Printf (printf)

------------------ ONE ODD DECIMAL DIGIT -----------------

oneOddDecimalDigit :: Int -> Bool
oneOddDecimalDigit =
(1 ==) . length . filter odd . digits

digits :: Int -> [Int]
digits = fmap (read . return) . show

--------------------------- TEST -------------------------
main :: IO ()
main = do
putStrLn "Below 1000:"
(putStrLn . table " " . chunksOf 10 . fmap show) \$
sampleBelow 1000

putStrLn "Count of matches below 10E6:"
(print . length) \$
sampleBelow 1000000

sampleBelow :: Int -> [Int]
sampleBelow =
filter oneOddDecimalDigit
. flip takeWhile primes
. (>)

------------------------- DISPLAY ------------------------

table :: String -> [[String]] -> String
table gap rows =
let ws = maximum . fmap length <\$> transpose rows
pw = printf . flip intercalate ["%", "s"] . show
in unlines \$ intercalate gap . zipWith pw ws <\$> rows
```
Output:
```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

Count of matches below 10E6:
2560```

## J

```   getOneOdds=. #~ 1 = (2 +/@:| "."0@":)"0
primesTo=. i.&.(p:inv)

_15 ]\ getOneOdds primesTo 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

# getOneOdds primesTo 1e6
2560
```

## jq

Works with: jq

Works with gojq, the Go implementation of jq

As noted in the Julia entry, if only one digit of a prime is odd, then that digit is in the ones place. The first solution presented here uses this observation to generate plausible candidates. The second solution is more brutish and slower but simpler.

See e.g. Erdős-primes#jq for a suitable implementation of `is_prime`.

### Fast solution

```### Preliminaries

def count(s): reduce s as \$x (null; .+1);

def emit_until(cond; stream):
label \$out | stream | if cond then break \$out else . end;

# Output: an unbounded stream
def primes_with_exactly_one_odd_digit:
# Output: a stream of candidate strings, in ascending numerical order
def candidates:
# input is \$width
def evens:
. as \$width
| if \$width == 0 then ""
else ("0","2","4","6","8") as \$i
| ((.-1)|evens) as \$j
| "\(\$i)\(\$j)"
end;
| evens as \$even
| ("1","3","5","7","9") as \$odd

(3,5,7),
(range(0; infinite)
| candidates | tonumber
| select(is_prime)) ;

emit_until(. > 1000; primes_with_exactly_one_odd_digit),

"\nThe number of primes less than 1000000 with exactly one odd digits is \(count(emit_until(. > 1000000; primes_with_exactly_one_odd_digit)))."```
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

The number of primes less than 1000000 with exactly one odd digits is 2560.
```

### A simpler but slower solution

```# Input is assumed to be prime.
# So we only need check the other digits are all even.
def prime_has_exactly_one_odd_digit:
if . == 2 then false
# The codepoints for the odd digits are also odd: [49,51,53,55,57]
else all(tostring | explode[:-1][];  . % 2 == 0)
end;

def primes_with_exactly_one_odd_digit_crudely:
# It is much faster to check for primality afterwards.
range(3; infinite; 2)
| select(prime_has_exactly_one_odd_digit and is_prime);```

## 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.

```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.")
```
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.
```

## Mathematica / Wolfram Language

```Labeled[Cases[
NestWhileList[NextPrime,
2, # <
1000 &], _?(Total[Mod[IntegerDigits@#, 2]] ==
1 &)], "Primes < 1000 with one odd digit", Top]
Labeled[Length@
Cases[NestWhileList[NextPrime,
2, # <
1000000 &], _?(Total[Mod[IntegerDigits@#, 2]] ==
1 &)], "Number of primes < 1,000,000 with one odd digit", Top]
```
Output:
```
Primes < 1000 with 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}
Number of primes < 1,000,000 with one odd digit
2560

```

## 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(1_000_000):
inc count
echo "\nFound \$# primes with only one odd digit below 1_000_000.".format(count)
```
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

```#!/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;
```
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.
Of course the last digit must/will be odd for all primes (other than 2 which has no odd digit anyway), and all digits prior to that must be even, eg 223 or 241, and not 257 or 743.

```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
```

## Quackery

`isprime` is defined at Primality by trial division#Quackery.

`evendigits` returns the nth number which has only even digits and ends with zero. There are self-evidently 5^5 of them less than 1000000.

We know that the only prime ending with a 5 is 5. So we can omit generating candidate numbers that end with a 5, and stuff the 5 into the right place in the nest (i.e. as item #1; item #0 will be 3) afterwards. This explains the lines `' [ 1 3 7 9 ] witheach` and `5 swap 1 stuff`.

```  [ [] swap
[ 5 /mod
rot join swap
dup 0 = until ]
drop
0 swap witheach
[ swap 10 * + ]
20 * ]                is evendigits ( n --> n )

[]
5 5 ** times
[ i^ evendigits
' [ 1 3 7 9 ] witheach
[ over +
dup isprime iff
[ swap dip join ]
else drop ]
drop ]
5 swap 1 stuff
dup say "Qualifying primes < 1000:"
dup findwith [ 999 > ] [ ]
split drop unbuild
1 split nip -1 split drop
nest\$ 44 wrap\$ cr cr
say "Number of qualifying primes < 1000000: "
size echo```
Output:
```Qualifying primes < 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

Number of qualifying primes < 1000000: 2560
```

## Raku

```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"
}
```
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

```/*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
```
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

```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```
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...
```

## RPL

Works with: HP version 49g
```≪
→STR 0
1 3 PICK SIZE FOR j
OVER j DUP SUB NUM 2 MOD + NEXT
NIP
≫ ≫ 'ODDCNT' STO

≪
{ } 1
DO
NEXTPRIME
IF DUP ODDCNT 1 == THEN SWAP OVER + SWAP END
UNTIL DUP 1000 ≥ END
DROP
```
Output:
```1: {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}
```

## Ruby

```require 'prime'

def single_odd?(pr)
d = pr.digits
d.first.odd? && d[1..].all?(&:even?)
end

res = Prime.each(1000).select {|pr| single_odd?(pr)}
res.each_slice(10){|s| puts "%4d"*s.size % s}

n = 1_000_000
count = Prime.each(n).count{|pr| single_odd?(pr)}
puts "\nFound #{count} single-odd-digit primes upto #{n}."
```
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 2560 single-odd-digit primes upto 1000000.
```

## Sidef

```func primes_with_one_odd_digit(upto, base = 10) {

upto = prev_prime(upto+1)

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.ilog(base)) {
even_digits.variations_with_repetition(k, {|*a|
next if (a.last == 0)
var v = a.digits2num(base)
odd_digits.each {|d|
var n = (v*base + d)
list << n if (n.is_prime && (n <= upto))
}
})
}

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}"
}
```
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

Library: Wren-math
Library: Wren-fmt
```import "./math" for Int
import "./fmt" for Fmt

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)
Fmt.tprint("\$,%(maxDigits)d", results, 9)
System.print("\nFound %(results.count) such primes.\n")

limit = 1e9 - 1
primes = Int.primeSieve(limit)
var count = 0
var pow = 10
for (prime in primes.skip(1)) {
if (Int.digits(prime)[0...-1].all { |d| d & 1 == 0 }) count = count + 1
if (prime > pow) {
Fmt.print("There are \$,7d such primes under \$,d", count, pow)
pow = pow * 10
}
}
Fmt.print("There are \$,7d such primes under \$,d", count, pow)
```
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       3 such primes under 10
There are      12 such primes under 100
There are      45 such primes under 1,000
There are     171 such primes under 10,000
There are     619 such primes under 100,000
There are   2,560 such primes under 1,000,000
There are  10,774 such primes under 10,000,000
There are  46,708 such primes under 100,000,000
There are 202,635 such primes under 1,000,000,000
```

## 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.
");
]```
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.
```