A005069 Sum of odd primes dividing n.

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, 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, 22, 29, 59, 8, 61, 31, 10, 0, 18, 14, 67, 17, 26, 12, 71, 3, 73, 37, 8, 19, 18, 16, 79

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

Primes counted without multiplicity. - Harvey P. Dale, Aug 28 2019

Links

• Antti Karttunen, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000265, A005066, A005067, A005068, A005078, A005082, A008472, A020639, A028234, A059841.

Programs

Mathematica

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 *)
Array[DivisorSum[#, # &, And[PrimeQ@ #, OddQ@ #] &] &, 79] (* Michael De Vlieger, Jul 11 2017 *)
Join[{0},Table[Total[FactorInteger[n][[All,1]]/.(2->0)],{n,2,100}]] (* Harvey P. Dale, Aug 28 2019 *)

PARI

a(n) = my(f=factor(n)); sum(k=1, #f~, if (((p=f[k,1])%2) == 1, p)); \\ Michel Marcus, Jul 11 2017

Scheme

(define (A005069 n) (cond ((= 1 n) 0) ((even? n) (A005069 (/ n 2))) (else (+ (A020639 n) (A005069 (A028234 n)))))) ;; Antti Karttunen, Jul 10 2017

Formula

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

G.f.: Sum_{k>=2} prime(k)*x^prime(k)/(1 - x^prime(k)). - Ilya Gutkovskiy, Dec 24 2016

From Antti Karttunen, Jul 10 & 11 2017: (Start)

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

a(n) = A008472(A000265(n)) = A008472(n) - 2*A059841(n).

a(n) = A005078(n) + A005082(n).

(End)

Extensions

More terms from Stefan Steinerberger, Jun 02 2007