A118503 Sum of digits of prime factors of n, with multiplicity.

0, 2, 3, 4, 5, 5, 7, 6, 6, 7, 2, 7, 4, 9, 8, 8, 8, 8, 10, 9, 10, 4, 5, 9, 10, 6, 9, 11, 11, 10, 4, 10, 5, 10, 12, 10, 10, 12, 7, 11, 5, 12, 7, 6, 11, 7, 11, 11, 14, 12, 11, 8, 8, 11, 7, 13, 13, 13, 14, 12, 7, 6, 13, 12, 9, 7, 13, 12, 8, 14, 8, 12, 10, 12, 13, 14, 9, 9, 16, 13

Offset: 1

Author

Jonathan Vos Post, May 06 2006

Keywords

• base
• easy
• nonn

Comments

This is to A095402 (Sum of digits of all distinct prime factors of n) as bigomega = A001222 is to omega = A001221. See also: A007953 Digital sum (i.e., sum of digits) of n.

Examples

			a(22) = 4 because 22 = 2 * 11 and the digital sum of 2 + the digital sum of 11 = 2 + 2 = 4.
a(121) = 4 because 121 = 11^2 = 11 * 11, summing the digits of the prime factors with multiplicity gives A007953(11) + A007953(11) = 2 + 2 = 4.
a(1000) = 21 because = 2^3 * 5^3 = 2 * 2 * 2 * 5 * 5 * 5 and 2 + 2 + 2 + 5 + 5 + 5 = 21, as opposed to A095402(1000) = 7.
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..10000 (terms 1..5000 from G. C. Greubel)

Crossrefs

Cf. A001221, A001222, A007953, A095402, A102217, A289142 (positions of multiples of 3's).

Programs

Maple

A118503 := proc(n) local a; a := 0 ; for p in ifactors(n)[2] do a := a+ op(2, p)*A007953(op(1, p)) ; end do: a ; end proc: # R. J. Mathar, Sep 14 2011

Mathematica

sdpf[n_]:=Total[Flatten[IntegerDigits/@Flatten[Table[#[[1]],{#[[2]]}]&/@FactorInteger[n]]]]; Join[{0},Array[sdpf,100,2]] (* Harvey P. Dale, Sep 19 2013 *)

PARI

A118503(n) = { my(f=factor(n)); sum(i=1, #f~, f[i, 2]*sumdigits(f[i, 1])); }; \\ Antti Karttunen, Jun 08 2024

Formula

a(n) = Sum_{i=1..k} (e_i)*A007953(p_i) where prime decomposition of n = (p_1)^(e_1) * (p_2)^(e_2) * ... * (p_k)^(e_k).

Extensions

a(0) removed by Joerg Arndt at the suggestion of Antti Karttunen, Jun 08 2024