A005066 Sum of squares of odd primes dividing n.

0, 0, 9, 0, 25, 9, 49, 0, 9, 25, 121, 9, 169, 49, 34, 0, 289, 9, 361, 25, 58, 121, 529, 9, 25, 169, 9, 49, 841, 34, 961, 0, 130, 289, 74, 9, 1369, 361, 178, 25, 1681, 58, 1849, 121, 34, 529, 2209, 9, 49, 25, 298, 169, 2809, 9, 146, 49, 370, 841, 3481, 34, 3721, 961, 58, 0, 194, 130, 4489, 289, 538, 74, 5041, 9, 5329, 1369

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn

Links

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

Crossrefs

Cf. A000265, A000290, A005063, A005067, A005068, A005069, A005079, A005083, A020639, A028234.

Programs

Mathematica

Table[Total[Select[Divisors[n],OddQ[#]&&PrimeQ[#]&]^2],{n,60}] (* Harvey P. Dale, May 02 2012 *)
Array[DivisorSum[#, #^2 &, And[PrimeQ@ #, OddQ@ #] &] &, 74] (* Michael De Vlieger, Jul 11 2017 *)
f[2, e_] := 0; f[p_, e_] := p^2; a[n_] := Plus @@ f @@@ FactorInteger[n]; a[1] = 0; Array[a, 100] (* Amiram Eldar, Jun 20 2022 *)

PARI

a(n) = sumdiv(n, d, ((d%2) && isprime(d))*d^2); \\ Michel Marcus, Jan 04 2017

Python

from sympy import primefactors
def a(n): return sum(p**2 for p in primefactors(n) if p % 2)
print([a(n) for n in range(1, 101)]) # Indranil Ghosh, Jul 11 2017

Scheme

(define (A005066 n) (cond ((= 1 n) 0) ((even? n) (A005066 (/ n 2))) (else (+ (A000290 (A020639 n)) (A005066 (A028234 n)))))) ;; Antti Karttunen, Jul 10 2017

Formula

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

G.f.: Sum_{k>=2} prime(k)^2*x^prime(k)/(1 - x^prime(k)). - Ilya Gutkovskiy, Jan 04 2017

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)^2 + a(A028234(n)).

a(n) = A005063(A000265(n)).

a(n) = A005079(n) + A005083(n).

(End)

Extensions

More terms from Antti Karttunen, Jul 10 2017