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A005067 Sum of cubes of odd primes dividing n.

%I A005067 #22 Jul 12 2017 05:25:14
%S A005067 0,0,27,0,125,27,343,0,27,125,1331,27,2197,343,152,0,4913,27,6859,125,
%T A005067 370,1331,12167,27,125,2197,27,343,24389,152,29791,0,1358,4913,468,27,
%U A005067 50653,6859,2224,125,68921,370,79507,1331,152,12167,103823,27,343,125,4940,2197,148877,27,1456,343,6886,24389,205379,152
%N A005067 Sum of cubes of odd primes dividing n.
%H A005067 Harvey P. Dale (terms 1 .. 1000) & Antti Karttunen, <a href="/A005067/b005067.txt">Table of n, a(n) for n = 1..10000</a>
%F A005067 Additive with a(p^e) = 0 if p = 2, p^3 otherwise.
%F A005067 G.f.: Sum_{k>=2} prime(k)^3*x^prime(k)/(1 - x^prime(k)). - _Ilya Gutkovskiy_, Jan 06 2017
%F A005067 From _Antti Karttunen_, Jul 10 2017: (Start)
%F A005067 a(1) = 0; after which, for even n: a(n) = a(n/2), for odd n: a(n) = A020639(n)^3 + a(A028234(n)).
%F A005067 a(n) = A005064(A000265(n)).
%F A005067 (End)
%t A005067 Join[{0},Table[Total[Select[Transpose[FactorInteger[n]][[1]],OddQ]^3],{n,2,50}]] (* _Harvey P. Dale_, Jun 09 2016 *)
%t A005067 Array[DivisorSum[#, #^3 &, And[PrimeQ@ #, OddQ@ #] &] &, 60] (* _Michael De Vlieger_, Jul 11 2017 *)
%o A005067 (Scheme) (define (A005067 n) (cond ((= 1 n) 0) ((even? n) (A005067 (/ n 2))) (else (+ (A000578 (A020639 n)) (A005067 (A028234 n)))))) ;; _Antti Karttunen_, Jul 10 2017
%o A005067 (PARI) a(n) = my(f=factor(n)); sum(k=1, #f~, if (((p=f[k,1])%2) == 1, p^3)); \\ _Michel Marcus_, Jul 11 2017
%Y A005067 Cf. A000265, A000578, A005064, A005066, A005068, A005069, A020639, A028234, A065091.
%K A005067 nonn
%O A005067 1,3
%A A005067 _N. J. A. Sloane_
%E A005067 More terms from _Antti Karttunen_, Jul 10 2017