Smarandache prime-digital sequence
The Smarandache prime-digital sequence (SPDS for brevity) is the sequence of primes whose digits are themselves prime.
For example 257 is an element of this sequence because it is prime itself and its digits: 2, 5 and 7 are also prime.
- Task
- Show the first 25 SPDS primes.
- Show the hundredth SPDS prime.
- See also
- OEIS A019546: Primes whose digits are primes.
- https://www.scribd.com/document/214851583/On-the-Smarandache-prime-digital-subsequence-sequences
F#
This task uses Extensible Prime Generator (F#) <lang fsharp> // Generate Smarandache prime-digital sequence. Nigel Galloway: May 31st., 2019 let rec spds g=seq{yield! g; yield! (spds (Seq.collect(fun g->[g*10+2;g*10+3;g*10+5;g*10+7]) g))}|>Seq.filter(isPrime) spds [2;3;5;7] |> Seq.take 25 |> Seq.iter(printfn "%d") printfn "\n\n100th item of this sequence is %d" (spds [2;3;5;7] |> Seq.item 99) printfn "1000th item of this sequence is %d" (spds [2;3;5;7] |> Seq.item 999) </lang>
- Output:
2 3 5 7 23 37 53 73 223 227 233 257 277 337 353 373 523 557 577 727 733 757 773 2237 2273 100th item of this sequence is 33223 1000th item of this sequence is 3273527
Go
As this task doesn't involve large numbers, a simple prime test routine is adequate. <lang go>package main
import "fmt"
func isPrime(n int) bool {
if n < 2 { return false } if n%2 == 0 { return n == 2 } if n%3 == 0 { return n == 3 } d := 5 for d*d <= n { if n%d == 0 { return false } d += 2 if n%d == 0 { return false } d += 4 } return true
}
func isSPDSPrime(n int) bool {
if !isPrime(n) { return false } for n > 0 { r := n % 10 if r != 2 && r != 3 && r != 5 && r != 7 { return false } n /= 10 } return true
}
func listSPDSPrimes(startFrom, countFrom, countTo int, printOne bool) int {
count := countFrom for n := startFrom; ; n += 2 { if isSPDSPrime(n) { count++ if !printOne { fmt.Printf("%2d. %d\n", count, n) } if count == countTo { if printOne { fmt.Printf("%2d. %d\n", count, n) } return n } } }
}
func main() {
fmt.Println("The first 25 terms of the Smarandache prime-digital sequence are:") fmt.Println(" 1. 2") n := listSPDSPrimes(3, 1, 25, false) fmt.Println("\nThe hundredth term of the sequence is:") listSPDSPrimes(n+2, 25, 100, true)
}</lang>
- Output:
The first 25 terms of the Smarandache prime-digital sequence are: 1. 2 2. 3 3. 5 4. 7 5. 23 6. 37 7. 53 8. 73 9. 223 10. 227 11. 233 12. 257 13. 277 14. 337 15. 353 16. 373 17. 523 18. 557 19. 577 20. 727 21. 733 22. 757 23. 773 24. 2237 25. 2273 The hundredth term of the sequence is: 100. 33223
Perl 6
<lang perl6># Implemented as a lazy, extendable list
my $spds = flat 2,3,5,7, (1..*).map: { grep { .is-prime }, [X~] |((2,3,5,7) xx $_), (3,7) };
say 'Smarandache prime-digitals:';
printf "%4d: %s\n", 1+$_, $spds[$_] for flat ^25, 99;</lang>
- Output:
Smarandache prime-digitals: 1: 2 2: 3 3: 5 4: 7 5: 23 6: 37 7: 53 8: 73 9: 223 10: 227 11: 233 12: 257 13: 277 14: 337 15: 353 16: 373 17: 523 18: 557 19: 577 20: 727 21: 733 22: 757 23: 773 24: 2237 25: 2273 100: 33223
REXX
The prime number generator has been simplified and very little optimization was included. <lang rexx>/*REXX program lists a sequence of SPDS (Smarandache prime-digital sequence) primes.*/ parse arg n m q /*get optional number of primes to find*/ if n== | n=="," then n= 25 /*Not specified? Then use the default.*/ if m== | m=="," then m= 100 /* " " " " " " */ say '═══listing the first' n "SPDS primes═══" call spds n say say '═══listing the last of ' m "SPDS primes═══" call spds -m
do i=1 to words(q); y=word(q, i); if y=="," then iterate say say '═══listing the last of ' y "SPDS primes═══" call spds -y end /*i*/
exit /*stick a fork in it, we're all done. */ /*──────────────────────────────────────────────────────────────────────────────────────*/ spds: parse arg x 1 ox; x= abs(x) /*obtain the limit to be used for list.*/
c= 0 /*C number of SPDS primes found so far*/ #= 0 /*# number of primes found so far*/ do j=1 by 2 while c<x; z= j /*start: 1st even prime, then use odd. */ if z==1 then z= 2 /*handle the even prime (special case) */ /* [↓] divide by the primes. ___ */ do k=2 to # while k*k<=z /*divide Z with all primes ≤ √ Z */ if z//@.k==0 then iterate j /*÷ by prev. prime? ¬prime ___ */ end /*j*/ /* [↑] only divide up to √ Z */ #= # + 1; @.#= z /*bump the prime count; assign prime #*/ if verify(z, 2357)>0 then iterate j /*Digits ¬prime? Then skip this prime.*/ c= c + 1 /*bump the number of SPDS primes found.*/ if ox<0 then iterate /*don't display it, display the last #.*/ say right(z, 21) /*maybe display this prime ──► terminal*/ end /*j*/ /* [↑] only display N number of primes*/ if ox<0 then say right(z, 21) /*display one (the last) SPDS prime. */ return</lang>
- output when using the default inputs:
═══listing the first 25 SPDS primes═══ 2 3 5 7 23 37 53 73 223 227 233 257 277 337 353 373 523 557 577 727 733 757 773 2237 2273 ═══listing the last of 100 SPDS primes═══ 33223 ═══listing the last of 1000 SPDS primes═══ 3273527
Ring
<lang ring>
- Project: Calmo primes
load "stdlib.ring" limit = 25 max = 300000 num = 0 see "working..." + nl see "wait for done..." + nl see "First 25 Calmo primes are:" + nl for n = 1 to max
if isprime(n) res = calmo(n) if res = 1 num = num + 1 if num < limit + 1 see "" + num + ". " + n + nl ok if num = 100 see "The hundredth Calmo prime is:" + nl see "" + num + ". " + n + nl exit ok ok ok
next see "done..." + nl
func calmo(p)
sp = string(p) for n = 1 to len(sp) if not isprime(sp[n]) return 0 ok next return 1
</lang>
- Output:
working... wait for done... First 25 Calmo primes are: 1. 2 2. 3 3. 5 4. 7 5. 23 6. 37 7. 53 8. 73 9. 223 10. 227 11. 233 12. 257 13. 277 14. 337 15. 353 16. 373 17. 523 18. 557 19. 577 20. 727 21. 733 22. 757 23. 773 24. 2237 25. 2273 The hundredth Calmo prime is: 100. 33223 done...
zkl
GNU Multiple Precision Arithmetic Library
Using GMP ( probabilistic primes), because it is easy and fast to generate primes.
Extensible prime generator#zkl could be used instead. <lang zkl>var [const] BI=Import("zklBigNum"); // libGMP
spds:=Walker.zero().tweak(fcn(ps){
var [const] nps=T(0,0,1,1,0,1,0,1,0,0); // 2,3,5,7 p:=ps.nextPrime().toInt(); if(p.split().filter( fcn(n){ 0==nps[n] }) ) return(Void.Skip); p // 733 --> (7,3,3) --> () --> good, 29 --> (2,9) --> (9) --> bad
}.fp(BI(1)));</lang> Or <lang zkl>spds:=Walker.zero().tweak(fcn(ps){
var [const] nps="014689".inCommon; p:=ps.nextPrime().toInt(); if(nps(p.toString())) return(Void.Skip); p // 733 --> "" --> good, 29 --> "9" --> bad
}.fp(BI(1)));</lang> <lang zkl>println("The first 25 terms of the Smarandache prime-digital sequence are:"); spds.walk(25).concat(",").println();
println("The hundredth term of the sequence is: ",spds.drop(100-25).value);</lang>
- Output:
The first 25 terms of the Smarandache prime-digital sequence are: 2,3,5,7,23,37,53,73,223,227,233,257,277,337,353,373,523,557,577,727,733,757,773,2237,2273 The hundredth term of the sequence is: 33223