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A005069 Sum of odd primes dividing n.

%I A005069 #26 Aug 28 2019 10:25:30
%S A005069 0,0,3,0,5,3,7,0,3,5,11,3,13,7,8,0,17,3,19,5,10,11,23,3,5,13,3,7,29,8,
%T A005069 31,0,14,17,12,3,37,19,16,5,41,10,43,11,8,23,47,3,7,5,20,13,53,3,16,7,
%U A005069 22,29,59,8,61,31,10,0,18,14,67,17,26,12,71,3,73,37,8,19,18,16,79
%N A005069 Sum of odd primes dividing n.
%C A005069 Primes counted without multiplicity. - _Harvey P. Dale_, Aug 28 2019
%H A005069 Antti Karttunen, <a href="/A005069/b005069.txt">Table of n, a(n) for n = 1..10000</a>
%F A005069 Additive with a(p^e) = 0 if p = 2, p otherwise.
%F A005069 G.f.: Sum_{k>=2} prime(k)*x^prime(k)/(1 - x^prime(k)). - _Ilya Gutkovskiy_, Dec 24 2016
%F A005069 From _Antti Karttunen_, Jul 10 & 11 2017: (Start)
%F A005069 a(1) = 0; after which, for even n: a(n) = a(n/2), for odd n: a(n) = A020639(n) + a(A028234(n)).
%F A005069 a(n) = A008472(A000265(n)) = A008472(n) - 2*A059841(n).
%F A005069 a(n) = A005078(n) + A005082(n).
%F A005069 (End)
%t A005069 a = {0, 0}; For[n = 3, n < 80, n++, su = 0; b = FactorInteger[n]; For[i = 1, i < Length[b] + 1, i++, If[OddQ[b[[i, 1]]], su = su + b[[i, 1]]]]; AppendTo[a, su]]; a (* _Stefan Steinerberger_, Jun 02 2007 *)
%t A005069 Array[DivisorSum[#, # &, And[PrimeQ@ #, OddQ@ #] &] &, 79] (* _Michael De Vlieger_, Jul 11 2017 *)
%t A005069 Join[{0},Table[Total[FactorInteger[n][[All,1]]/.(2->0)],{n,2,100}]] (* _Harvey P. Dale_, Aug 28 2019 *)
%o A005069 (Scheme) (define (A005069 n) (cond ((= 1 n) 0) ((even? n) (A005069 (/ n 2))) (else (+ (A020639 n) (A005069 (A028234 n)))))) ;; _Antti Karttunen_, Jul 10 2017
%o A005069 (PARI) a(n) = my(f=factor(n)); sum(k=1, #f~, if (((p=f[k,1])%2) == 1, p)); \\ _Michel Marcus_, Jul 11 2017
%Y A005069 Cf. A000265, A005066, A005067, A005068, A005078, A005082, A008472, A020639, A028234, A059841.
%K A005069 nonn,easy
%O A005069 1,3
%A A005069 _N. J. A. Sloane_
%E A005069 More terms from _Stefan Steinerberger_, Jun 02 2007