A060209 Dunckley sequence: number of bases in which the n-th composite number is a Smith number.

0, 0, 0, 0, 1, 0, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 3, 1, 2, 2, 1, 3, 1, 2, 1, 1, 1, 4, 1, 2, 3, 1, 2, 2, 2, 3, 1, 4, 1, 3, 3, 5, 1, 4, 3, 1, 3, 1, 1, 5, 6, 2, 2, 1, 1, 7, 1, 2, 2, 4, 6, 1, 2, 1, 2, 1, 2, 1, 4, 1, 1, 2, 2, 2, 5, 3, 7, 3, 2, 4, 1, 1, 6, 3, 1, 4, 2, 3, 2, 3, 1, 1, 1, 5, 2, 4, 1, 5, 5, 1, 3, 2, 1, 5, 3, 2

Offset: 1

Author

Alfred and Dominic Vella (dunckley(AT)thevellas.freeserve.co.uk), Mar 19 2001

Keywords

• nonn
• base

Examples

			The first 4 composite numbers, 4, 6, 8, and 9, are not Smith numbers in any base, so a(n) = 0 for n = 1 to 4.
A002808(5) = 10 is a Smith number in one base, 4, so a(5) = 1.
		

References

• A. Vella and D. Vella, On Smith and Dunckley Numbers, Mathematics Today (Bull. Inst. Math. Appl), Vol. 37, No. 2 (2001), 54-56.
• A. Vella and D. Vella, More Properties of Dunckley Numbers (in preparation).

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A002808, A006753.

Programs

Mathematica

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]]; smithCount /@ Select[Range[100], CompositeQ] (* Amiram Eldar, Aug 21 2020 *)

Extensions

a(1) added and offset corrected by Amiram Eldar, Aug 21 2020