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

1,2

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.

Links

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

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).

Example

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.

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 *)

Prog

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

Crossrefs

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

Sequence in context: A106492 A338038 A112264 * A086295 A360269 A345423

Adjacent sequences: A118500 A118501 A118502 * A118504 A118505 A118506

Keyword

base,easy,nonn

Author

Jonathan Vos Post, May 06 2006

Extensions

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

Status

approved