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

1,3

Links

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

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)

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 *)

Prog

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

Crossrefs

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

Sequence in context: A067153 A057405 A167354 * A076262 A167301 A177741

Adjacent sequences: A005063 A005064 A005065 * A005067 A005068 A005069

Keyword

nonn

Author

N. J. A. Sloane

Extensions

More terms from Antti Karttunen, Jul 10 2017

Status

approved