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%I #30 Dec 29 2023 10:56:56
%S 13,14,16,17,20,30,31,32,34,35,37,38,50,70,71,73,74,76,79,91,92,95,97,
%T 98,110,113,118,119,131,133,199,311,337,373,733,772,775,778,779,919,
%U 991,1118,3337,7771,77779
%N Look at the set of numbers obtained by permuting the digits of n in all possible ways, then remove n itself from the set. If the remaining numbers are all primes, then n is in the sequence.
%C If it exists, a(46) > 5*10^11. - _Lars Blomberg_, Mar 31 2018
%e 119 is in the sequence because every permutation of its digits excluding 119 (i.e., 191 and 911) is a prime.
%e 11 is not in the sequence, because when 11 is removed from the set, no numbers are left.
%p lis := [];
%p for n from 1 to 10000 do
%p nn := convert(n, base, 10);
%p pp := combinat[permute](nn);
%p if nops(pp) = 1 then
%p next
%p end if;
%p lOk := true;
%p for p in pp do
%p if p = nn then
%p next: #exclude n
%p end if;
%p if `not`(isprime(convert(p, base, 10, 10^nops(p))[])) then
%p lOk := false; break
%p end if
%p end do;
%p if lOk then
%p lis := [op(lis), n]
%p end if
%p end do:
%p lis := lis;
%t rnapQ[n_]:=Module[{p=Rest[FromDigits/@Permutations[IntegerDigits[ n]]]},If[ Length[p]==0, False, AllTrue[p,PrimeQ]]]; Select[Range[80000],rnapQ] (* The program uses the AllTrue function from Mathematica version 10 *) (* _Harvey P. Dale_, Jan 24 2019 *)
%o (PARI) isok(n) = {v = []; d = digits(n); for (k=0, (#d)!-1, p = numtoperm(#d, k); dp = vector(#d, j, d[p[j]]); np = subst(Pol(dp), x, 10); v = Set(concat(v, np));); v = setminus(v, Set(n)); if (#v == 0, return (0)); for (k=1, #v, if (!isprime(v[k]), return (0));); return (1);} \\ _Michel Marcus_, Apr 18 2016
%o (Python)
%o from sympy import isprime
%o from itertools import count, islice, permutations
%o def agen(): yield from (k for k in count(1) if len(set(s:=str(k)))!=1 and all((t:=int("".join(m)))==k or isprime(t) for m in permutations(s)))
%o print(list(islice(agen(), 45))) # _Michael S. Branicky_, Dec 29 2023
%Y Cf. A095179, A111347.
%Y Cf. A003459. - _Altug Alkan_, Apr 18 2016
%K nonn,base,more
%O 1,1
%A _César Eliud Lozada_, Apr 18 2016
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