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%I #6 Feb 11 2021 22:58:39
%S 1,3,6,8,12,16,24,36,48,72,96,120,144,216,288,432,576,720,864,1080,
%T 1152,1440,2016,2160,2880,3600,4320,5040,5760,6480,7200,8640,10080,
%U 10800,14400,20160,21600,25920,28800,32400,40320,43200,50400,64800,86400,100800,129600
%N Numbers k that are divisible by their product of digits in a record number of bases 1 < b < k.
%C The corresponding record values are 0, 1, 2, 3, 5, 6, 9, 12, 15, 20, 22, 23, 32, 35, 46, 52, 58, 68, 69, 71, 76, 95, 96, 106, 126, 137, 145, 149, 161, 164, 185, 191, 196, 218, 249, 266, 286, 290, 310, 318, 330, 375, 387, 428, 471, 510, 564, ...
%C It seems that most terms are least integers of a prime signature (A025487), but some are not: e.g., 3 and 2016.
%e The values of A341434(k) for k=1..8 are 0, 0, 1, 1, 1, 2, 2, 3. The record values, 0, 1, 2 and 3, occur at 1, 3, 6 and 8, the first 4 terms of this sequence.
%t q[n_, b_] := (p = Times @@ IntegerDigits[n, b]) > 0 && Divisible[n, p]; a[n_] := Count[Range[2, n], _?(q[n, #] &)]; s = {}; am = -1; Do[a1 = a[n]; If[a1 > am, am = a1; AppendTo[s, n]], {n, 1, 10^3}]; s
%Y Indices of records in A341434.
%Y Cf. A330813.
%K nonn,base
%O 1,2
%A _Amiram Eldar_, Feb 11 2021