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A331565 The base 10 numbers with a digit product > 0 and which when written in bases 3,4,5,6,7,8,9 have two or more other base representations with the same digit product.

%I A331565 #18 Jan 24 2020 16:33:58
%S A331565 1,2,3,4,5,6,7,91,491,921,1138,1234,4853,13581,23568,29242,42161,
%T A331565 42162,42163,42164,42991,43365,44313,83342,83651,85226,114382,153881,
%U A331565 155462,159422,232868,291862,296183,352486,372642,398543,419563,441194,465326,616146,625431,625523,635813
%N A331565 The base 10 numbers with a digit product > 0 and which when written in bases 3,4,5,6,7,8,9 have two or more other base representations with the same digit product.
%C A331565 For terms 10 < a(n) < 10^9 none have a base-3 representation whose digit product equals the base-10 product. The first such entry using the base-4 representation is 491.
%e A331565 6 is a term as 6_10 = 6_7 = 6_8 = 6_9, so it has three other base representations where the digit product also equals 6.
%e A331565 91 is a term as 91_10 = 331_5 = 133_8, so it has two other base representations where the digit product also equals 9.
%e A331565 491 is a term as 491_10 = 13223_4 = 3431_5, so it has two other base representations where the digit product also equals 36.
%t A331565 proDig[n_, b_] := Times @@ IntegerDigits[n, b]; seqQ[n_] := Module[{prod = proDig[n, 10], count = 0}, If[prod > 0, Do[If[proDig[n, b] == prod, count++]; If[count == 2, Break[]], {b, 3, 9}]]; count == 2]; Select[Range[650000], seqQ] (* _Amiram Eldar_, Jan 21 2020 *)
%o A331565 (PARI) isok(n) = {my(p=vecprod(digits(n))); (p != 0) && (sum(k=3, 9, p==vecprod(digits(n,k))) >= 2);} \\ _Michel Marcus_, Jan 21 2020
%Y A331565 Cf. A007954, A007089, A007090, A007091, A007092, A007093, A007094, A007095, A331561.
%Y A331565 Subsequence of A052382 (zeroless numbers).
%K A331565 nonn,base
%O A331565 1,2
%A A331565 _Scott R. Shannon_, Jan 20 2020