A005067 Sum of cubes of odd primes dividing n.

0, 0, 27, 0, 125, 27, 343, 0, 27, 125, 1331, 27, 2197, 343, 152, 0, 4913, 27, 6859, 125, 370, 1331, 12167, 27, 125, 2197, 27, 343, 24389, 152, 29791, 0, 1358, 4913, 468, 27, 50653, 6859, 2224, 125, 68921, 370, 79507, 1331, 152, 12167, 103823, 27, 343, 125, 4940, 2197, 148877, 27, 1456, 343, 6886, 24389, 205379, 152

Offset

1,3

Links

Harvey P. Dale (terms 1 .. 1000) & Antti Karttunen, Table of n, a(n) for n = 1..10000

Formula

Additive with a(p^e) = 0 if p = 2, p^3 otherwise.

G.f.: Sum_{k>=2} prime(k)^3*x^prime(k)/(1 - x^prime(k)). - Ilya Gutkovskiy, Jan 06 2017

From Antti Karttunen, Jul 10 2017: (Start)

a(1) = 0; after which, for even n: a(n) = a(n/2), for odd n: a(n) = A020639(n)^3 + a(A028234(n)).

a(n) = A005064(A000265(n)).

(End)

Mathematica

Join[{0}, Table[Total[Select[Transpose[FactorInteger[n]][[1]], OddQ]^3], {n, 2, 50}]] (* Harvey P. Dale, Jun 09 2016 *)
Array[DivisorSum[#, #^3 &, And[PrimeQ@ #, OddQ@ #] &] &, 60] (* Michael De Vlieger, Jul 11 2017 *)

Prog

(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
(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

Crossrefs

Cf. A000265, A000578, A005064, A005066, A005068, A005069, A020639, A028234, A065091.

Sequence in context: A277042 A224118 A178737 * A163574 A076108 A040735

Adjacent sequences: A005064 A005065 A005066 * A005068 A005069 A005070

Keyword

nonn

Author

N. J. A. Sloane

Extensions

More terms from Antti Karttunen, Jul 10 2017

Status

approved