A330813 -id:A330813 - cp's OEIS frontend

A337294 Composite numbers k that are Smith numbers in a record number of bases 1 < b <= k.

4, 10, 15, 27, 42, 60, 72, 78, 174, 204, 222, 378, 438, 663, 1352, 1446, 2022, 2526, 2598, 3462, 4038, 4542, 6054, 12102, 22182, 30336, 35432, 39318, 44358, 55446, 72582, 90726, 99798, 110886, 120966, 157254, 181446, 235878, 288294, 332646, 399174, 432438, 665286

Offset: 1

Amiram Eldar, Aug 21 2020

Values of A002808 at the indices of records of A060209.

The corresponding number of bases are 0, 1, 2, 3, 4, 5, 6, 7, 9, 10, 14, 15, 19, 20, 21, 22, 27, 29, 31, 33, 35, 40, 48, 59, 66, 67, 71, 76, 80, 81, 88, 97, 98, 101, 105, 118, 119, 130, 131, 152, 156, 167, 187, ...

			a(1) = 4 since it is the least composite number and it is not a Smith number in any base 1 < b <= 4.
a(2) = 10 since it is the least number that is a Smith number in any base 1 < b <= 10: 10 = 2 * 5 is, 22_4 = 2_4 * 11_4 in base 4, and 2 + 2 = 2 + (1 + 1) = 4.

Cf. A002808, A006753, A060209.

Similar sequences: A107129, A330813.

digSum[n_, b_] := Plus @@ IntegerDigits[n, b]; smithCount[n_] := If[! CompositeQ[n], 0, Module[{c = 0, f = FactorInteger[n]}, p = f[[;; , 1]]; e = f[[;; , 2]]; Do[If[Total[e*(digSum[#, b] & /@ p)] == digSum[n, b], c++], {b, 2, n}]; c]]; seq = {}; cmax = -1; Do[If[CompositeQ[n] && (c = smithCount[n]) > cmax, cmax = c; AppendTo[seq, n]], {n, 1, 666}]; seq

A341435 Numbers k that are divisible by their product of digits in a record number of bases 1 < b < k.

Original entry on oeis.org

1, 3, 6, 8, 12, 16, 24, 36, 48, 72, 96, 120, 144, 216, 288, 432, 576, 720, 864, 1080, 1152, 1440, 2016, 2160, 2880, 3600, 4320, 5040, 5760, 6480, 7200, 8640, 10080, 10800, 14400, 20160, 21600, 25920, 28800, 32400, 40320, 43200, 50400, 64800, 86400, 100800, 129600

Offset: 1

Amiram Eldar, Feb 11 2021

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, ...

It seems that most terms are least integers of a prime signature (A025487), but some are not: e.g., 3 and 2016.

			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.

Indices of records in A341434.

Cf. A330813.

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

cp's OEIS Frontend

A337294 Composite numbers k that are Smith numbers in a record number of bases 1 < b <= k.

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