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Composite numbers k that are Smith numbers in a record number of bases 1 < b <= k.
0

%I #9 Nov 07 2024 15:20:29

%S 4,10,15,27,42,60,72,78,174,204,222,378,438,663,1352,1446,2022,2526,

%T 2598,3462,4038,4542,6054,12102,22182,30336,35432,39318,44358,55446,

%U 72582,90726,99798,110886,120966,157254,181446,235878,288294,332646,399174,432438,665286

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

%C Values of A002808 at the indices of records of A060209.

%C 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, ...

%e a(1) = 4 since it is the least composite number and it is not a Smith number in any base 1 < b <= 4.

%e 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.

%t 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

%Y Cf. A002808, A006753, A060209.

%Y Similar sequences: A107129, A330813.

%K nonn,base

%O 1,1

%A _Amiram Eldar_, Aug 21 2020