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

1,8

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

Example

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.

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 *)

Crossrefs

Cf. A002808, A006753.

Sequence in context: A238744 A030421 A085021 * A037830 A174353 A319582

Adjacent sequences: A060206 A060207 A060208 * A060210 A060211 A060212

Keyword

nonn,base

Author

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

Extensions

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

Status

approved