A130680 Numbers n such that n = (a_1 + a_2 + ... + a_p)*(a_1^3 + a_2^3 + ... + a_p^3), where n has the decimal expansion a_1a_2...a_p.

1, 1215, 3700, 11680, 13608, 87949

Offset: 1

Author

Yalcin Aktar, Jun 29 2007

Keywords

• base
• fini
• full
• nonn

Comments

This sequence is finite and all the terms are listed. Proof: Let a_1a_2...a_p be the decimal expansion of n. Then p <= log_10(n)+1. Furthermore we have a_i <= 9, therefore (a_1 + a_2 + ... + a_p) <= 9*(log_10(n)+1) and (a_1^3 + a_2^3 + ... + a_p^3) <= 9^3*(log_10(n)+1). On the other hand, for all n > 300000 we have 9^4*(log_10(n)+1)^2 < n. A computer search confirms that we indeed have found all terms.

Examples

			87949 = (8+7+9+4+9)*(8^3+7^3+9^3+4^3+9^3).
		

Crossrefs

Cf. A115518.

Programs

Mathematica

For[n = 1, n < 1000000, n++, b = IntegerDigits[n]; If[Sum[b[[i]], {i, 1, Length[b]}] * Sum[b[[i]]^3, {i, 1, Length[b]}] == n, Print[n]]]
ffQ[n_]:=Module[{c=IntegerDigits[n]},Total[c]Total[c^3]==n]; Select[ Range[ 90000],ffQ] (* Harvey P. Dale, Oct 18 2013 *)

Extensions

Edited by Stefan Steinerberger, Jul 13 2007