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

Author

N. J. A. Sloane

Keywords

• nonn

Links

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

Crossrefs

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

Programs

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 *)

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

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

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)

Extensions

More terms from Antti Karttunen, Jul 10 2017