cp's OEIS Frontend

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

%I A118503 #28 Jun 08 2024 21:00:02
%S A118503 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,
%T A118503 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,
%U A118503 13,14,12,7,6,13,12,9,7,13,12,8,14,8,12,10,12,13,14,9,9,16,13
%N A118503 Sum of digits of prime factors of n, with multiplicity.
%C A118503 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.
%H A118503 Antti Karttunen, <a href="/A118503/b118503.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..5000 from G. C. Greubel)
%F A118503 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).
%e A118503 a(22) = 4 because 22 = 2 * 11 and the digital sum of 2 + the digital sum of 11 = 2 + 2 = 4.
%e A118503 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.
%e A118503 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.
%p A118503 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
%t A118503 sdpf[n_]:=Total[Flatten[IntegerDigits/@Flatten[Table[#[[1]],{#[[2]]}]&/@FactorInteger[n]]]]; Join[{0},Array[sdpf,100,2]] (* _Harvey P. Dale_, Sep 19 2013 *)
%o A118503 (PARI) A118503(n) = { my(f=factor(n)); sum(i=1, #f~, f[i, 2]*sumdigits(f[i, 1])); }; \\ _Antti Karttunen_, Jun 08 2024
%Y A118503 Cf. A001221, A001222, A007953, A095402, A102217, A289142 (positions of multiples of 3's).
%K A118503 base,easy,nonn
%O A118503 1,2
%A A118503 _Jonathan Vos Post_, May 06 2006
%E A118503 a(0) removed by _Joerg Arndt_ at the suggestion of _Antti Karttunen_, Jun 08 2024