Prime numbers which contain 123

Prime numbers which contain 123 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.

Task

Find those primes   n   whose decimal representation contains the consecutive digits   123,   where   n   <   100,000.

Stretch goal

As above, but only show the count of those primes   n   that contain the (above) string,   where   n   <   1,000,000.

11l

F is_prime(a)
   I a == 2
      R 1B
   I a < 2 | a % 2 == 0
      R 0B
   L(i) (3 .. Int(sqrt(a))).step(2)
      I a % i == 0
         R 0B
   R 1B

F is_prime123(n)
   R ‘123’ C String(n) & is_prime(n)

V c = 0
L(n) 100'000
   I is_prime123(n)
      c++
      print(‘#5’.format(n), end' I c % 8 == 0 {"\n"} E ‘ ’)
print()
print(‘Found ’c‘ "123" primes less than 100000’)
c = 0
L(n) 1'000'000
   I is_prime123(n)
      c++
print()
print(‘Found ’c‘ "123" primes less than 1000000’)

Output:

 1123  1231  1237  8123 11239 12301 12323 12329
12343 12347 12373 12377 12379 12391 17123 20123
22123 28123 29123 31123 31231 31237 34123 37123
40123 41231 41233 44123 47123 49123 50123 51239
56123 59123 61231 64123 65123 70123 71233 71237
76123 81233 81239 89123 91237 98123 
Found 46 "123" primes less than 100000

Found 451 "123" primes less than 1000000

ALGOL 68

Library: ALGOL 68-primes

BEGIN  # find primes whose decimal representation contains 123 #
    INT max prime = 1 000 000;
    # sieve the primes to max prime #
    PR read "primes.incl.a68" PR
    []BOOL prime = PRIMESIEVE max prime;
    # find the appropriate primes #
    # as observed by the Wren sample, the primes must have a least 4 digits #
    INT show max    = 100 000;
    INT p123 count := 0;
    FOR n FROM 1001 TO UPB prime DO
        IF prime[ n ] THEN
            # have a prime #
            BOOL has 123 := FALSE;
            INT  v       := n;
            WHILE v >= 123 AND NOT ( has 123 := v MOD 1000 = 123 ) DO
                v OVERAB 10
            OD;
            IF has 123 THEN
                # the prime contains "123" #
                p123 count +:= 1;
                IF n <= show max THEN
                    print( ( whole( n, -7 ) ) );
                    IF p123 count MOD 12 = 0 THEN print( ( newline ) ) FI
                FI
            FI
        FI;
        IF n = 100 000 THEN
            print( ( newline, "Found ", whole( p123 count, 0 ), " ""123"" primes below ", whole( show max, 0 ), newline ) )
        FI
    OD;
    print( ( newline, "Found ", whole( p123 count, 0 ), " ""123"" primes below ", whole( UPB prime, 0 ), newline ) )
END

Output:

   1123   1231   1237   8123  11239  12301  12323  12329  12343  12347  12373  12377
  12379  12391  17123  20123  22123  28123  29123  31123  31231  31237  34123  37123
  40123  41231  41233  44123  47123  49123  50123  51239  56123  59123  61231  64123
  65123  70123  71233  71237  76123  81233  81239  89123  91237  98123
Found 46 "123" primes below 100000

Found 451 "123" primes below 1000000

Arturo

upTo100K: select select 2..99999 => odd? => prime?
upTo1M: upTo100K ++ select select 100001..999999 => odd? => prime?

contains123?: function [x] -> contains? to :string x "123"

loop split.every:10 select upTo100K => contains123? 'a ->
    print map a => [pad to :string & 5]

print ""
print ["'123' Numbers < 1000000:" size select upTo1M => contains123?]

Output:

 1123  1231  1237  8123 11239 12301 12323 12329 12343 12347 
12373 12377 12379 12391 17123 20123 22123 28123 29123 31123 
31231 31237 34123 37123 40123 41231 41233 44123 47123 49123 
50123 51239 56123 59123 61231 64123 65123 70123 71233 71237 
76123 81233 81239 89123 91237 98123 

'123' Numbers < 1000000: 451

AWK

# syntax: GAWK -f PRIME_NUMBERS_WHICH_CONTAIN_123.AWK
BEGIN {
    start = 1
    stop = 99999
    for (i=start; i<=stop; i++) {
      if (is_prime(i) && i ~ /123/) {
        printf("%6d%1s",i,++count%10?"":"\n")
      }
    }
    printf("\nPrimes with '123' %d-%d: %d\n",start,stop,count)
    stop = 999999
    for (i=100000; i<=stop; i++) {
      if (is_prime(i) && i ~ /123/) {
        count++
      }
    }
    printf("\nPrimes with '123' %d-%d: %d\n",start,stop,count)
    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:

  1123   1231   1237   8123  11239  12301  12323  12329  12343  12347
 12373  12377  12379  12391  17123  20123  22123  28123  29123  31123
 31231  31237  34123  37123  40123  41231  41233  44123  47123  49123
 50123  51239  56123  59123  61231  64123  65123  70123  71233  71237
 76123  81233  81239  89123  91237  98123
Primes with '123' 1-99999: 46

Primes with '123' 1-999999: 451

BASIC256

global columna
print "Prime numbers which contain 123"
print
limite = 100000
call prime(limite, true)
print : print
print "Found "; columna; " prime numbers below "; limite
limite = 1e6
call prime(limite, false)
print : print
print "Found "; columna; " prime numbers below "; int(limite)
end

function isPrime(v)
    if v < 2 then return False
    if v mod 2 = 0 then return v = 2
    if v mod 3 = 0 then return v = 3
    d = 5
    while d * d <= v
        if v mod d = 0 then return False else d += 2
    end while
    return True
end function

subroutine prime(limite, mostrar)
    columna = 0

    for n = 1 to limite
        strn$ = string(n)
        if isPrime(n) and instr(strn$, "123") > 0 then
            columna += 1
            if mostrar then
                print rjust(string(n), 7);
                if columna mod 8 = 0 then print
            end if
        endif
    next n
end subroutine

Output:

Same as FreeBASIC entry.

C++

#include <iostream>
#include <string>
#include <vector>
#include <cmath>

bool isPrime( int number ) {
   if ( number < 2 ) {
      return false ;
   }
   int stop =  std::sqrt( static_cast<double>( number ) ) ;
   for ( int i = 2 ; i <= stop ; ++i )
      if ( number % i == 0 )
         return false ;
   return true ;
}

bool condition( int n ) {
   std::string numberstring { std::to_string( n ) } ;
   return isPrime( n ) && numberstring.find( "123" ) != std::string::npos ;
}

int main( ) {
   std::vector<int> wantedPrimes ; 
   for ( int i = 1 ; i < 100000 ; i++ ) {
      if ( condition( i ) ) 
         wantedPrimes.push_back( i ) ;
   }
   int count = 0 ;
   for ( int i : wantedPrimes ) {
      std::cout << i << ' ' ;
      count++ ;
      if ( count % 10 == 0 ) {
         std::cout << std::endl ;
      }
   }
   count = wantedPrimes.size( ) ;
   for ( int i = wantedPrimes.back( ) + 1 ; i < 1000000 ; i++ ) {
      if ( condition ( i ) ) 
         count++ ;
   }
   std::cout << std::endl ;
   std::cout << "There are " << count << " such numbers below 1000000!\n" ;
   return 0 ;
}

Output:

1123 1231 1237 8123 11239 12301 12323 12329 12343 12347 
12373 12377 12379 12391 17123 20123 22123 28123 29123 31123 
31231 31237 34123 37123 40123 41231 41233 44123 47123 49123 
50123 51239 56123 59123 61231 64123 65123 70123 71233 71237 
76123 81233 81239 89123 91237 98123 
There are 451 such numbers below 1000000!

Clojure

(ns primes-found-app
  (:require [clojure.string :as str])
  (:gen-class))

(defn is-prime? [n]
  (if (< 1 n)
    (empty? (filter #(= 0 (mod n %)) (range 2 n)))
    false))

(defn get-prime-numbers [n]
  (filter is-prime? (take n (range))))

(defn numbers-to-str [xs]
  (map #(str %) xs))

(defn is-includes-123? [s]
  (str/includes? s "123"))

(defn solution [number]
  (->>
   (get-prime-numbers number)
   (numbers-to-str)
   (filter is-includes-123?)))

(defn main []
  (let [result (count (solution 1000000))]
  (prn (solution 100000))
  (format "There are %d primes that contain '123' below 1000000." result)
  ))

(main)

Output:

("1123" "1231" "1237" "8123" "11239" "12301" "12323" "12329" "12343" "12347" "12373" "12377" "12379" "12391" "17123" "20123" "22123" "28123" "29123" "31123" "31231" "31237" "34123" "37123" "40123" "41231" "41233" "44123" "47123" "49123" "50123" "51239" "56123" "59123" "61231" "64123" "65123" "70123" "71233" "71237" "76123" "81233" "81239" "89123" "91237" "98123")
"There are 451 primes that contain '123' below 1000000."

CLU

isqrt = proc (s: int) returns (int)
    x0: int := s/2
    if x0=0 then return(s) end
    x1: int := (x0 + s/x0)/2
    while x1 < x0 do
        x0 := x1
        x1 := (x0 + s/x0)/2
    end
    return(x0)
end isqrt

sieve = proc (n: int) returns (array[bool])
    prime: array[bool] := array[bool]$fill(2,n-1,true)
    for p: int in int$from_to(2,isqrt(n)) do
        if prime[p] then
            for c: int in int$from_to_by(p*p,n,p) do
                prime[c] := false
            end
        end
    end
    return(prime)
end sieve

start_up = proc ()
    po: stream := stream$primary_output()
    count: int := 0
    prime: array[bool] := sieve(1000000)
    
    for p: int in array[bool]$indexes(prime) do
        if ~prime[p] then continue end
        if string$indexs("123", int$unparse(p))=0 then continue end
        count := count + 1
        if p < 100000 then 
            stream$putright(po, int$unparse(p), 7) 
            if count//10=0 then stream$putl(po, "") end
        end
    end
    
    stream$putl(po, "\nThere are " || int$unparse(count)
                || " primes that contain '123' below 1,000,000.")
end start_up

Output:

   1123   1231   1237   8123  11239  12301  12323  12329  12343  12347
  12373  12377  12379  12391  17123  20123  22123  28123  29123  31123
  31231  31237  34123  37123  40123  41231  41233  44123  47123  49123
  50123  51239  56123  59123  61231  64123  65123  70123  71233  71237
  76123  81233  81239  89123  91237  98123
There are 451 primes that contain '123' below 1,000,000.

Delphi

Works with: Delphi version 6.0

Library: SysUtils,StdCtrls

procedure ShowPrimesWith123(Memo: TMemo);
var N,Sum,Cnt1,Cnt2: integer;
var NS,S: string;
begin
Cnt1:=0;
Cnt2:=0;
Sum:=0;
for N:=123 to 1000000-1 do
 if IsPrime(N) then
	begin
	NS:=IntToStr(N);
	if Pos('123',NS)>0 then
		begin
		Inc(Cnt1);
		if N<100000 then
			begin
			Inc(Cnt2);
			S:=S+Format('%6d',[N]);
			If (Cnt2 mod 8)=0 then S:=S+CRLF;
			end;
		end;
	end;
Memo.Lines.Add(S);
Memo.Lines.Add('Count <   100,000 = '+IntToStr(Cnt2));
Memo.Lines.Add('Count < 1,000,000 = '+IntToStr(Cnt1));
end;

Output:

  1123  1231  1237  8123 11239 12301 12323 12329
 12343 12347 12373 12377 12379 12391 17123 20123
 22123 28123 29123 31123 31231 31237 34123 37123
 40123 41231 41233 44123 47123 49123 50123 51239
 56123 59123 61231 64123 65123 70123 71233 71237
 76123 81233 81239 89123 91237 98123
Count <   100,000 = 46
Count < 1,000,000 = 451
Elapsed Time: 147.996 ms.

F#

This task uses Extensible Prime Generator (F#).

// Numbers containing 123. Nigel Galloway: July 14th., 2021
let rec fN g=if g%1000=123 then true else if g<1230 then false else fN(g/10)
primes32()|>Seq.takeWhile((>)100000)|>Seq.filter fN|>Seq.iter(printf "%d "); printfn ""
printfn "Count to 1 million is %d" (primes32()|>Seq.takeWhile((>)1000000)|>Seq.filter fN|>Seq.length)

Output:

1123 1231 1237 8123 11239 12301 12323 12329 12343 12347 12373 12377 12379 12391 17123 20123 22123 28123 29123 31123 31231 31237 34123 37123 40123 41231 41233 44123 47123 49123 50123 51239 56123 59123 61231 64123 65123 70123 71233 71237 76123 81233 81239 89123 91237 98123
Count to 1 million is 451

Factor

Works with: Factor version 0.99 2021-06-02

USING: assocs formatting grouping io kernel literals math
math.functions math.functions.integer-logs math.primes
math.statistics sequences sequences.extras sequences.product
sorting tools.memory.private tools.time ;

<<
CONSTANT: d { 0 1 2 3 4 5 6 7 8 9 }  ! digits that can be anywhere
CONSTANT: e { 1 3 7 9 }              ! digits that can be at the end
>>

CONSTANT: digits {
    ${ { 1 } { 2 } { 3 } d d d d d e }
    ${ d { 1 } { 2 } { 3 } d d d d e }
    ${ d d { 1 } { 2 } { 3 } d d d e }
    ${ d d d { 1 } { 2 } { 3 } d d e }
    ${ d d d d { 1 } { 2 } { 3 } d e }
    ${ d d d d d { 1 } { 2 } { 3 } e }
    ${ d d d d d d { 1 } { 2 } { 3 } }
}

: candidates ( -- seq )
    digits [ <product-sequence> ] map-concat
    [ <reversed> 0 [ 10^ * + ] reduce-index ] map ;

: 123primes ( -- assoc )
    candidates [ prime? ] filter
    [ integer-log10 1 + ] collect-by >alist natural-sort ;

[
    "Decimal primes under 100,000 which contain '123':" print
    123primes dup [ 4 of ] [ 5 of append ] bi natural-sort
    10 group [ [ commas "%8s" printf ] each nl ] each nl
    [ [ 10^ commas ] [ length ] bi* ] assoc-map
    unzip cum-sum [ commas ] map swap zip
    [ "Found %7s such primes under %s.\n" printf ] assoc-each
] time

Output:

Decimal primes under 100,000 which contain '123':
   1,123   1,231   1,237   8,123  11,239  12,301  12,323  12,329  12,343  12,347
  12,373  12,377  12,379  12,391  17,123  20,123  22,123  28,123  29,123  31,123
  31,231  31,237  34,123  37,123  40,123  41,231  41,233  44,123  47,123  49,123
  50,123  51,239  56,123  59,123  61,231  64,123  65,123  70,123  71,233  71,237
  76,123  81,233  81,239  89,123  91,237  98,123

Found       4 such primes under 10,000.
Found      46 such primes under 100,000.
Found     451 such primes under 1,000,000.
Found   4,412 such primes under 10,000,000.
Found  43,548 such primes under 100,000,000.
Found 435,853 such primes under 1,000,000,000.
Running time: 21.938685601 seconds

FreeBASIC

Dim Shared As Integer column

Function isPrime(Byval ValorEval As Integer) As Boolean
    If ValorEval <= 1 Then Return False
    For i As Integer = 2 To Int(Sqr(ValorEval))
        If ValorEval Mod i = 0 Then Return False
    Next i
    Return True
End Function

Sub prime(limite As Long, mostrar As Boolean)
    column = 0
    
    For n As Integer = 1 To limite
        Dim As String strn = Str(n)
        If isPrime(n) And Instr(strn,"123") > 0 Then
            column += 1
            If mostrar Then 
                Print Using "  ##### "; n;
                If (column Mod 8) = 0 Then Print
            End If
        End If
    Next n
End Sub

Print !"N£meros primos que contienen 123:\n"
Dim As Long limite = 1e5
prime(limite, true)
Print !"\n\n\Encontrados "; column; " n£meros primos por debajo de"; limite
limite = 1e6
prime(limite, false)
Print !"\n\n\Encontrados "; column; " n£meros primos por debajo de"; limite
Sleep

Output:

Números primos que contienen 123:

   1123    1231    1237    8123   11239   12301   12323   12329
  12343   12347   12373   12377   12379   12391   17123   20123
  22123   28123   29123   31123   31231   31237   34123   37123
  40123   41231   41233   44123   47123   49123   50123   51239
  56123   59123   61231   64123   65123   70123   71233   71237
  76123   81233   81239   89123   91237   98123

Encontrados  46 números primos por debajo de 100000


Encontrados  451 números primos por debajo de 1000000

FutureBasic

include "NSLog.incl"

local fn IsPrime( n as long ) as BOOL
  long i
  BOOL result = YES
  
  if ( n < 2 ) then result = NO : exit fn
  for i = 2 to n + 1
    if ( i * i <= n ) and ( n mod i == 0 )
      result = NO : exit fn
    end if
  next
end fn = result


local fn PrimeWith123( limit as long )
  long        i, column = 1
  CFStringRef numStr
  
  NSLog( @"Prime numbers less than 100,000 which contain '123':\n" )
  for i = 1 to limit
    numStr = fn StringWithFormat( @"%lu", i )
    if ( fn IsPrime( i ) ) and ( fn StringContainsString( numStr, @"123" ) )
      NSLog( @"%-6lu\b", i )
      if column == 10 then column = 0 : NSLog( @"" )
      column++
    end if
  next
end fn

fn PrimeWith123( 100000 )

HandleEvents

Output:

Prime numbers less than 100,000 which contain '123':

12373 12377 12379 12391 17123 20123 22123 28123 29123 31123 
31231 31237 34123 37123 40123 41231 41233 44123 47123 49123 
50123 51239 56123 59123 61231 64123 65123 70123 71233 71237 
76123 81233 81239 89123 91237 98123 

Go

Translation of: Wren

Library: Go-rcu

package main

import (
    "fmt"
    "rcu"
    "strings"
)

func main() {
    limit := 100_000
    primes := rcu.Primes(limit * 10)
    var results []int
    for _, p := range primes {
        if p < 1000 || p > 99999 {
            continue
        }
        ps := fmt.Sprintf("%s", p)
        if strings.Contains(ps, "123") {
            results = append(results, p)
        }
    }
    climit := rcu.Commatize(limit)
    fmt.Printf("Primes under %s which contain '123' when expressed in decimal:\n", climit)
    for i, p := range results {
        fmt.Printf("%7s ", rcu.Commatize(p))
        if (i+1)%10 == 0 {
            fmt.Println()
        }
    }
    fmt.Println("\n\nFound", len(results), "such primes under", climit, "\b.")

    limit = 1_000_000
    climit = rcu.Commatize(limit)
    count := len(results)
    for _, p := range primes {
        if p < 100_000 {
            continue
        }
        ps := fmt.Sprintf("%s", p)
        if strings.Contains(ps, "123") {
            count++
        }
    }
    fmt.Println("\nFound", count, "such primes under", climit, "\b.")
}

Output:

Primes under 100,000 which contain '123' when expressed in decimal:
  1,123   1,231   1,237   8,123  11,239  12,301  12,323  12,329  12,343  12,347 
 12,373  12,377  12,379  12,391  17,123  20,123  22,123  28,123  29,123  31,123 
 31,231  31,237  34,123  37,123  40,123  41,231  41,233  44,123  47,123  49,123 
 50,123  51,239  56,123  59,123  61,231  64,123  65,123  70,123  71,233  71,237 
 76,123  81,233  81,239  89,123  91,237  98,123 

Found 46 such primes under 100,000.

Found 451 such primes under 1,000,000.

Haskell

import Data.List ( isInfixOf )

isPrime :: Int -> Bool
isPrime n 
   |n < 2 = False
   |otherwise = null $ filter (\i -> mod n i == 0 ) [2 .. root]
   where
      root :: Int
      root = floor $ sqrt $ fromIntegral n
   
condition :: Int -> Bool
condition n = isPrime n && isInfixOf "123" ( show n )

solution :: [Int]
solution = filter condition [2..99999]

Output:

[1123,1231,1237,8123,11239,12301,12323,12329,12343,12347,12373,12377,12379,12391,17123,20123,22123,28123,29123,31123,31231,31237,34123,37123,40123,41231,41233,44123,47123,49123,50123,51239,56123,59123,61231,64123,65123,70123,71233,71237,76123,81233,81239,89123,91237,98123]

J

   p:I.1 2 3 +./@E."1/ 10 #.inv p:i.p:inv 1e5 NB. primes less than 1e5 containing decimal digit sequence 1 2 3
1123 1231 1237 8123 11239 12301 12323 12329 12343 12347 12373 12377 12379 12391 17123 20123 22123 28123 29123 31123 31231 31237 34123 37123 40123 41231 41233 44123 47123 49123 50123 51239 56123 59123 61231 64123 65123 70123 71233 71237 76123 81233 81239 89123 91237 98123
   +/1 2 3 +./@E."1/ 10 #.inv p:i.p:inv 1e6 NB. count of primes less than 1e6 containing decimal digit sequence 1 2 3
451

p:inv 1e5 is 9592 -- the number of primes less than 1e5. p:i.9592 enumerates those primes (first is 2, last is 99991). 10 #.inv converts this to the corresponding table of decimal digits (0 0 0 0 2 for the first row, 9 9 9 9 1 for the last row). 1 2 3 +./@E."1 identifies those rows containing the sequence 1 2 3, and I. gets the indices of those rows and, finally, p: gets the prime numbers corresponding to those indices. Similarly, +/ counts the rows with suitable primes.

jq

Works with: jq

Works with gojq, the Go implementation of jq

For a suitable implementation of `is_prime`, see e.g. # Erdős-primes#jq.

def count(stream): reduce stream as $i (0; .+1);

def digits: tostring | explode;

# Input: an upper bound, or `infinite`
def primes_with_123:
  ("123"| digits) as $d123
  | range(123; .; 2))
  | select( (digits | index($d123)) and is_prime);

100000 | primes_with_123,

(1000000
 | "\nThere are \(count(primes_with_123)) \"123\" primes less than \(.).")

Output:

(Abbreviated)

1123
1231
1237
8123
...
81233
81239
89123
91237
98123

There are 451 "123" primes less than 1000000.

Julia

using Primes

function containstringinbase(N, str, base, verbose = true)
    arr = filter(n -> occursin(str, string(n, base=base)), primes(N))
    println("\n\nFound $(length(arr)) primes < $N which contain the string $str in base $base representation.")
    verbose && foreach(p -> print(rpad(p[2], 6), p[1] % 12 == 0 ? "\n" : ""), enumerate(arr))
end

containstringinbase(100_000, "123", 10)
containstringinbase(1_000_000, "123", 10, false)
containstringinbase(1_000_000_000, "123", 10, false)

Output:

Found 46 primes < 100000 which contain the string 123 in base 10 representation.
1123  1231  1237  8123  11239 12301 12323 12329 12343 12347 12373 12377 
12379 12391 17123 20123 22123 28123 29123 31123 31231 31237 34123 37123
40123 41231 41233 44123 47123 49123 50123 51239 56123 59123 61231 64123
65123 70123 71233 71237 76123 81233 81239 89123 91237 98123

Found 451 primes < 1000000 which contain the string 123 in base 10 representation.


Found 435002 primes < 1000000000 which contain the string 123 in base 10 representation.

Ksh

#!/bin/ksh

# Prime numbers which contain 123

#	# Variables:
#
integer MAX_SHOW=100000 MAX_COUNT=1000000 primecnt=0
pattrn='123'
typeset -a parr

#	# Functions:
#

#	# Function _isprime(n) return 1 for prime, 0 for not prime
#
function _isprime {
	typeset _n ; integer _n=$1
	typeset _i ; integer _i

	(( _n < 2 )) && return 0
	for (( _i=2 ; _i*_i<=_n ; _i++ )); do
		(( ! ( _n % _i ) )) && return 0
	done
	return 1
}


 ######
# main #
 ######

for ((i=2; i<MAX_COUNT; i++)); do
	_isprime ${i}
	if (( $? )); then
		if [[ ${i} == *${pattrn}* ]]; then
			((primecnt++))
			(( i < MAX_SHOW )) && parr+=( ${i} )
		fi
	fi
done

print ${parr[*]}
print ${#parr[*]} found under $MAX_SHOW
echo ; print ${primecnt} found under $MAX_COUNT

Output:

1123 1231 1237 8123 11239 12301 12323 12329 12343 12347 12373 12377 12379
12391 17123 20123 22123 28123 29123 31123 31231 31237 34123 37123 40123 41231 41233 44123 47123
49123 50123 51239 56123 59123 61231 64123 65123 70123 71233 71237 76123 81233 81239 89123 91237 98123
46 found under 100000

451 found under 1000000

Mathematica/Wolfram Language

s1 = Select[Prime[Range[PrimePi[10^5]]], IntegerDigits/*(SequenceCount[#, {1, 2, 3}] &)/*GreaterThan[0]]
Length[s1]

Length[Select[Prime[Range[PrimePi[10^6]]], IntegerDigits/*(SequenceCount[#, {1, 2, 3}] &)/*GreaterThan[0]]]

Output:

{1123, 1231, 1237, 8123, 11239, 12301, 12323, 12329, 12343, 12347, 12373, 12377, 12379, 12391, 17123, 20123, 22123, 28123, 29123, 31123, 31231, 31237, 34123, 37123, 40123, 41231, 41233, 44123, 47123, 49123, 50123, 51239, 56123, 59123, 61231, 64123, 65123, 70123, 71233, 71237, 76123, 81233, 81239, 89123, 91237, 98123}
46
451

Nim

import sequtils, strutils

const N = 1_000_000 - 1   # Sieve of Erathostenes size.

# Sieve of Erathostenes.
var composite: array[2..N, bool]

for n in countup(3, N, 2):  # We ignore the even values.
  let n2 = n * n
  if n2 > N: break
  if not composite[n]:
    for k in countup(n2, N, 2 * n):
      composite[k] = true

template isPrime(n: Positive): bool = not composite[n]


iterator primes123(lim: Positive): int =
  var n = 1001    # First odd value with four digits.
  while n <= lim:
    if n.isPrime and ($n).find("123") >= 0:
      yield n
    inc n, 2


let list = toSeq(primes123(100_000 - 1))
echo "Found ", list.len, " “123” primes less than 100_000:"
for i, n in list:
  stdout.write ($n).align(5), if (i + 1) mod 8 == 0: '\n' else: ' '
echo '\n'

var count = 0
for _ in primes123(1_000_000): inc count
echo "Found ", count, " “123” primes less than 1_000_000."

Output:

Found 46 “123” primes less than 100_000:
 1123  1231  1237  8123 11239 12301 12323 12329
12343 12347 12373 12377 12379 12391 17123 20123
22123 28123 29123 31123 31231 31237 34123 37123
40123 41231 41233 44123 47123 49123 50123 51239
56123 59123 61231 64123 65123 70123 71233 71237
76123 81233 81239 89123 91237 98123 

Found 451 “123” primes less than 1_000_000.

Perl

Library: ntheory

#!/usr/bin/perl

use strict;
use warnings;
use ntheory qw( primes );

my @hundredthousand = grep /123/, @{ primes(1e5) };
my $million = grep /123/, @{ primes(1e6) };
print "@hundredthousand\n\nmillion count is $million\n" =~ s/.{70}\K /\n/gr;

Output:

1123 1231 1237 8123 11239 12301 12323 12329 12343 12347 12373 12377 12379
12391 17123 20123 22123 28123 29123 31123 31231 31237 34123 37123 40123
41231 41233 44123 47123 49123 50123 51239 56123 59123 61231 64123 65123
70123 71233 71237 76123 81233 81239 89123 91237 98123

million count is 451

Phix

with javascript_semantics
function m123(string s) return match("123",s) end function
function fn(integer n) return filter(apply(get_primes_le(n),sprint),m123) end function
sequence res = fn(100_000)
printf(1,"found %d < 100_000: %s\n",{length(res),join(shorten(res,"",5))})
printf(1,"found %d < 1_000_000\n",{length(fn(1_000_000))})

Output:

found 46 < 100_000: 1123 1231 1237 8123 11239 ... 81233 81239 89123 91237 98123
found 451 < 1_000_000

PureBasic

Translation of: FreeBASIC

Procedure isPrime(v.i)
  If     v <= 1    : ProcedureReturn #False
  ElseIf v < 4     : ProcedureReturn #True
  ElseIf v % 2 = 0 : ProcedureReturn #False
  ElseIf v < 9     : ProcedureReturn #True
  ElseIf v % 3 = 0 : ProcedureReturn #False
  Else
    Protected r = Round(Sqr(v), #PB_Round_Down)
    Protected f = 5
    While f <= r
      If v % f = 0 Or v % (f + 2) = 0
        ProcedureReturn #False
      EndIf
      f + 6
    Wend
  EndIf
  ProcedureReturn #True
EndProcedure

Global column.i

Procedure prime(limite.l, mostrar.b)
    column = 0
    
    For n = 1 To limite
        strn.s = Str(n)
        If isPrime(n) And FindString(strn,"123") > 0:
            column + 1
            If mostrar:
                Print(FormatNumber(n,0,"","") + "  ")
                If column % 8 = 0: PrintN("") : EndIf
            EndIf
        EndIf
    Next n
EndProcedure

OpenConsole()
PrintN("Prime numbers which contain 123")
limite.l = 1e5
prime(limite, #True)
PrintN(#CRLF$ + #CRLF$ + "Found " + Str(column) + " prime numbers below " + Str(limite))
limite = 1e6
prime(limite, #False)
PrintN(#CRLF$ + "Found " + Str(column) + " prime numbers below " + Str(limite))
Input()
CloseConsole()

Output:

Same as FreeBASIC entry.

Python

Translation of: FreeBASIC

#!/usr/bin/python

def prime(limite, mostrar):
    global columna
    columna = 0
    
    for n in range(limite):
        strn = str(n)
        if isPrime(n) and ('123' in str(n)):
            columna += 1                
            if mostrar == True:
                print(n, end="  ");
                if columna % 8 == 0:
                    print('')
    return columna


if __name__ == "__main__":
    print("Números primos que contienen 123:")
    limite = 100000
    prime(limite, True)
    print("\n\nEncontrados ", columna, " números primos por debajo de", limite)
    limite = 1000000
    prime(limite, False)
    print("\n\nEncontrados ", columna, " números primos por debajo de", limite)

Output:

Igual que la entrada de FreeBASIC.

Quackery

eratosthenes and isprime are defined at Sieve of Eratosthenes#Quackery.

  100000 eratosthenes

  [ false swap
    [ dup 122 > while
      dup 1000 mod
      123 = iff
        [ dip not ]
        done
      10 / again ]
    drop ]            is contains123 ( n --> b )

  []
  100000 times
    [ i^ isprime if
        [ i^ contains123 if
            [ i^ join ] ] ]
  [] swap witheach
     [ number$ nested join ]
  48 wrap$

Output:

1123 1231 1237 8123 11239 12301 12323 12329
12343 12347 12373 12377 12379 12391 17123 20123
22123 28123 29123 31123 31231 31237 34123 37123
40123 41231 41233 44123 47123 49123 50123 51239
56123 59123 61231 64123 65123 70123 71233 71237
76123 81233 81239 89123 91237 98123

Raku

my @p123 = ^∞ .grep: { (.contains: 123) && .is-prime };

put display @p123[^(@p123.first: * > 1e5, :k)];

put "\nCount up to 1e6: ", ~ +@p123[^(@p123.first: * > 1e6, :k)];

sub display ($list, :$cols = 10, :$fmt = '%6d', :$title = "{+$list} matching:\n" )   {
    cache $list;
    $title ~ $list.batch($cols)».fmt($fmt).join: "\n"
}

Output:

46 matching:
  1123   1231   1237   8123  11239  12301  12323  12329  12343  12347
 12373  12377  12379  12391  17123  20123  22123  28123  29123  31123
 31231  31237  34123  37123  40123  41231  41233  44123  47123  49123
 50123  51239  56123  59123  61231  64123  65123  70123  71233  71237
 76123  81233  81239  89123  91237  98123

Count up to 1e6: 451

REXX

This REXX versions allows the user to specify   (on the command line)   the high limit for the primes to be searched,  
the number of columns to be shown,   and the decimal string that the primes must contain.   A negative number for
the number of columns suppresses the list of primes,   but shows the total number of primes found.

/*REXX program finds & displays primes (in decimal) that contain the decimal digits 123.*/
parse arg  hi cols str .                         /*obtain optional argument from the CL.*/
if   hi=='' |   hi==","  then   hi= 100000       /*Not specified?  Then use the default.*/
if cols=='' | cols==","  then cols=     10       /* "      "         "   "   "     "    */
if  str=='' |  str==","  then  str=    123       /* "      "         "   "   "     "    */
call genP                                        /*build array of semaphores for primes.*/
w= 10                                            /*width of a number in any column.     */
title= ' primes  N  (in decimal) that contain the decimal digits string '     str      ,
                                             " (in order),  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=1  for #                               /*search list of primes that have a str*/
     if pos(str, @.j)==0   then iterate          /*does this decimal prime contain "123"*/          /* ◄■■■■■■■ the filter.*/
     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 decimal) that contain the decimal digits string  123  (in order),  where  N  <  100,000
───────┼───────────────────────────────────────────────────────────────────────────────────────────────────────────────
   1   │      1,123      1,231      1,237      8,123     11,239     12,301     12,323     12,329     12,343     12,347
  11   │     12,373     12,377     12,379     12,391     17,123     20,123     22,123     28,123     29,123     31,123
  21   │     31,231     31,237     34,123     37,123     40,123     41,231     41,233     44,123     47,123     49,123
  31   │     50,123     51,239     56,123     59,123     61,231     64,123     65,123     70,123     71,233     71,237
  41   │     76,123     81,233     81,239     89,123     91,237     98,123
───────┴───────────────────────────────────────────────────────────────────────────────────────────────────────────────

Found  46  primes  N  (in decimal) that contain the decimal digits string  123  (in order),  where  N  <  100,000

output   when using the default inputs:     1000000   -1

Found  451  primes  N  (in decimal) that contain the decimal digits string  123  (in order),  where  N  <  1,000,000

Ring

load "stdlib.ring"
row = 0

see "working..." + nl
see "Prime numbers which contain 123 are:" + nl

for n = 1 to 100000
    strn = string(n)
    ind = substr(strn,"123")
    if isprime(n) and ind > 0
       see "" + n + " "
       row++
       if row%5 = 0
          see nl
       ok
    ok  
next

see nl + "Found " + row + " numbers" + nl
see "done..." + nl

Output:

working...
Prime numbers which contain 123 are:
1123 1231 1237 8123 11239 
12301 12323 12329 12343 12347 
12373 12377 12379 12391 17123 
20123 22123 28123 29123 31123 
31231 31237 34123 37123 40123 
41231 41233 44123 47123 49123 
50123 51239 56123 59123 61231 
64123 65123 70123 71233 71237 
76123 81233 81239 89123 91237 
98123 
Found 46 numbers
done...

RPL

Brute force

≪ ALOG → max
  ≪ { } 1123 
     WHILE DUP max < REPEAT
       NEXTPRIME
       IF DUP →STR "123" POS THEN SWAP OVER + SWAP END
     END NIP
≫ ≫ 'TASK’ STO

Output:

1: { 1123 1231 1237 8123 11239 12301 12323 12329 12343 12347 12373 12377 12379 12391 17123 20123 22123 28123 29123 31123 31231 31237 34123 37123 40123 41231 41233 44123 47123 49123 50123 51239 56123 59123 61231 64123 65123 70123 71233 71237 76123 81233 81239 89123 91237 98123 }

Runs in 12 minutes 45 seconds on a HP-50g

Lexicographic approach

RPL code	Comment
≪ 3 - DUP ALOG 1 - { } → ndigits maxleft results ≪ 0 DO DUP DUP "" IFTE ndigits OVER SIZE - ALOG 1 + DUP 2 x 3 - FOR j DUP "123" + j →STR TAIL + STR→ IF DUP ISPRIME? THEN 'results' STO+ ELSE DROP END 2 STEP DROP 1 + UNTIL DUP maxleft > END DROP results ≫ ≫ 'PR123N’ STO	PR123N ( n_digits → { primes_with_123 } ) n_digits -= 3 ; maxleft = 10^n_digits - 1 leftval = 0 loop left = leftval == 0 ? "" ; leftval tmp = 10^(n_digits - length(leftval)) + 1 for j = tmp to (tmp*2 - 3) step 2 n = left + "123" + j[2..] add n to results if prime next j leftval++ end loop display results (unsorted)

4 PR123N 5 PR123N + SORT

Runs in 22 seconds on a HP-50g (35 times faster than brute force) - same results as above. Finding the 451 primes under one million takes 4 minutes 30 seconds.

Ruby

require 'prime'

RE = /123/
puts Prime.each(100_000).select {|prime| RE.match? prime.to_s}.join(" "), ""
puts "#{Prime.each(1_000_000).count {|prime| RE.match? prime.to_s} } 123-primes below 1 million."

Output:

1123 1231 1237 8123 11239 12301 12323 12329 12343 12347 12373 12377 12379 12391 17123 20123 22123 28123 29123 31123 31231 31237 34123 37123 40123 41231 41233 44123 47123 49123 50123 51239 56123 59123 61231 64123 65123 70123 71233 71237 76123 81233 81239 89123 91237 98123

451 123-primes below 1 million.

Sidef

func numbers_with_subdigits(upto, base = 10, s = 123.digits(base)) {
    Enumerator({|callback|
        for k in (0 .. base**(upto.len(base) - s.len)) {

            var d = k.digits(base)

            for i in (0 .. d.len) {
                var n = d.clone.insert(i, s...).digits2num(base)
                callback(n) if (n <= upto)
            }

            var z = d.clone.insert(d.len, s...)
            loop {
                var n = z.insert(d.len, 0).digits2num(base)
                (n <= upto) ? callback(n) : break
            }
        }
    })
}

say "Decimal primes under 100,000 which contain '123':"
numbers_with_subdigits(1e5).grep { .is_prime }.sort.each_slice(10, {|*a|
    say a.map { '%6s' % _ }.join(' ')
})

say ''

for n in (4..8) {
    var count = numbers_with_subdigits(10**n).grep { .is_prime }.len
    say "Found #{'%6s' % count.commify} such primes < 10^#{n}"
}

Output:

Decimal primes under 100,000 which contain '123':
  1123   1231   1237   8123  11239  12301  12323  12329  12343  12347
 12373  12377  12379  12391  17123  20123  22123  28123  29123  31123
 31231  31237  34123  37123  40123  41231  41233  44123  47123  49123
 50123  51239  56123  59123  61231  64123  65123  70123  71233  71237
 76123  81233  81239  89123  91237  98123

Found      4 such primes < 10^4
Found     46 such primes < 10^5
Found    451 such primes < 10^6
Found  4,412 such primes < 10^7
Found 43,548 such primes < 10^8

Wren

Library: Wren-math

Library: Wren-fmt

Library: Wren-seq

The only number under 1,000 which can possibly satisfy the task description is 123 and that's clearly divisible by 3 and hence composite.

import "/math" for Int
import "/fmt" for Fmt
import "/seq" for Lst

var limit = 1e5
var primes = Int.primeSieve(limit * 10).where { |p| p > 999 }
var results = primes.where { |p| p < limit && p.toString.contains("123") }.toList
Fmt.print("Primes under $,d which contain '123' when expressed in decimal:", limit)
for (chunk in Lst.chunks(results, 10)) Fmt.print("$,7d", chunk)
Fmt.print("\nFound $,d such primes under $,d.", results.count, limit)

limit = 1e6
var count = primes.count { |p| p.toString.contains("123") }
Fmt.print("\nFound $,d such primes under $,d.", count, limit)

Output:

Primes under 100,000 which contain '123' when expressed in decimal:
  1,123   1,231   1,237   8,123  11,239  12,301  12,323  12,329  12,343  12,347
 12,373  12,377  12,379  12,391  17,123  20,123  22,123  28,123  29,123  31,123
 31,231  31,237  34,123  37,123  40,123  41,231  41,233  44,123  47,123  49,123
 50,123  51,239  56,123  59,123  61,231  64,123  65,123  70,123  71,233  71,237
 76,123  81,233  81,239  89,123  91,237  98,123

Found 46 such primes under 100,000.

Found 451 such primes under 1,000,000.

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;
];

func Has123(N);         \Return 'true' if N contains sequential digits 1 2 3
int  N, C, D;
[C:= 3;
repeat  N:= N/10;
        D:= rem(0);
        if D = C then
                [C:= C-1;
                if C = 0 then return true;
                ]
        else    [C:= 3;
                if D = C then C:= 2;
                ];
until   N=0;
return false;
];

int N, Count;
[Count:= 0;
for N:= 123 to 1_000_000-1 do
    [if Has123(N) then if IsPrime(N) then
        [Count:= Count+1;
        if N < 100_000 then
           [IntOut(0, N);
           if rem(Count/10) = 0 then CrLf(0) else ChOut(0, 9\tab\);
           ];
        ];
    if N = 100_000 then
        [CrLf(0);  IntOut(0, Count);  Text(0, " ^"123^" primes found below 100,000.")];
    ];
CrLf(0);  IntOut(0, Count);  Text(0, " ^"123^" primes found below 1,000,000.");
]

Output:

1123    1231    1237    8123    11239   12301   12323   12329   12343   12347
12373   12377   12379   12391   17123   20123   22123   28123   29123   31123
31231   31237   34123   37123   40123   41231   41233   44123   47123   49123
50123   51239   56123   59123   61231   64123   65123   70123   71233   71237
76123   81233   81239   89123   91237   98123   
46 "123" primes found below 100,000.
451 "123" primes found below 1,000,000.

Yabasic

Translation of: FreeBASIC

sub isPrime(v)
    if v < 2 then return False : fi
    if mod(v, 2) = 0 then return v = 2 : fi
    if mod(v, 3) = 0 then return v = 3 : fi
    d = 5
    while d * d <= v
        if mod(v, d) = 0 then return False else d = d + 2 : fi
    wend
    return True
end sub

sub prime(limite, mostrar)
    local n
	n = 0
    columna = 0
    
    for n = 1 to limite
        strn$ = str$(n)
        if isPrime(n) and instr(strn$,"123") > 0 then
            columna = columna + 1
            if mostrar then 
                print "  ", n using "#####", " ";
                if mod(columna, 8) = 0 then print : fi
            endif
        endif
    next n
end sub

print "N£meros primos que contienen 123:\n"
limite = 1e5
prime(limite, true)
print "\n\nEncontrados ", columna, " n£meros primos por debajo de ", limite
limite = 1e6
prime(limite, false)
print "\n\nEncontrados ", columna, " n£meros primos por debajo de ", limite
end

Output:

Igual que la entrada de FreeBASIC.