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Smallest prime whose product of digits is 7^n.
6

%I #7 Nov 07 2021 09:46:29

%S 11,7,11177,1777,71777,1777717,1177717771,77777177,7177717777,

%T 1777777777,71777777777,1717777777777,7177777777777,17777777777777,

%U 17177777777777717,7717777777777777,1177777777177777777,1777777777777777177,7777177777777777777

%N Smallest prime whose product of digits is 7^n.

%H Michael S. Branicky, <a href="/A090841/b090841.txt">Table of n, a(n) for n = 0..999</a>

%e a(6) = 1177717771 because its digital product is 7^6, and it is prime.

%p a:= proc(n) local k, t; for k from 0 do t:= min(select(isprime,

%p map(x-> parse(cat(x[])), combinat[permute]([1$k, 7$n]))));

%p if t<infinity then return t fi od

%p end:

%p seq(a(n), n=0..18); # _Alois P. Heinz_, Nov 07 2021

%t NextPrim[n_] := Block[{k = n + 1}, While[ !PrimeQ[k], k++ ]; k]; a = Table[0, {18}]; p = 2; Do[q = Log[7, Times @@ IntegerDigits[p]]; If[q != 0 && IntegerQ[q] && a[[q]] == 0, a[[q]] = p; Print[q, " = ", p]]; p = NextPrim[p], {n, 1, 10^9}]

%t For a(8); a = Map[ FromDigits, Permutations[{1, 1, 7, 7, 7, 7, 7, 7, 7, 7}]]; Min[ Select[a, PrimeQ[ # ] &]]

%o (Python)

%o from sympy import isprime

%o from sympy.utilities.iterables import multiset_permutations as mp

%o def a(n):

%o if n < 2: return [11, 7][n]

%o digits = n

%o while True:

%o for p in mp("1"*(digits-n) + "7"*n, digits):

%o t = int("".join(p))

%o if isprime(t): return t

%o digits += 1

%o print([a(n) for n in range(19)]) # _Michael S. Branicky_, Nov 07 2021

%Y Cf. A089365, A088653, A090840, A091465, A089298.

%K base,nonn

%O 0,1

%A _Robert G. Wilson v_, Dec 09 2003

%E a(17) and beyond from _Michael S. Branicky_, Nov 07 2021