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.

1, 2, 3, 4, 5, 6, 7, 91, 491, 921, 1138, 1234, 4853, 13581, 23568, 29242, 42161, 42162, 42163, 42164, 42991, 43365, 44313, 83342, 83651, 85226, 114382, 153881, 155462, 159422, 232868, 291862, 296183, 352486, 372642, 398543, 419563, 441194, 465326, 616146, 625431, 625523, 635813

Offset: 1

Author

Scott R. Shannon, Jan 20 2020

Keywords

• nonn
• base

Comments

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.

Examples

			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.
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.
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.
		

Crossrefs

Cf. A007954, A007089, A007090, A007091, A007092, A007093, A007094, A007095, A331561.

Subsequence of A052382 (zeroless numbers).

Programs

Mathematica

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 *)

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