A005068 Sum of 4th powers of odd primes dividing n.

0, 0, 81, 0, 625, 81, 2401, 0, 81, 625, 14641, 81, 28561, 2401, 706, 0, 83521, 81, 130321, 625, 2482, 14641, 279841, 81, 625, 28561, 81, 2401, 707281, 706, 923521, 0, 14722, 83521, 3026, 81, 1874161, 130321, 28642, 625, 2825761, 2482, 3418801, 14641, 706, 279841, 4879681, 81, 2401, 625, 83602, 28561, 7890481, 81

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn

Links

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

Crossrefs

Cf. A000265, A000583, A005065, A005066, A005067, A005069, A020639, A028234.

Programs

Mathematica

Array[DivisorSum[#, #^4 &, And[PrimeQ@ #, OddQ@ #] &] &, 54] (* Michael De Vlieger, Jul 11 2017 *)
f[2, e_] := 0; f[p_, e_] := p^4; a[n_] := Plus @@ f @@@ FactorInteger[n]; a[1] = 0; Array[a, 50] (* Amiram Eldar, Jun 20 2022 *)

PARI

a(n) = my(f=factor(n)); sum(k=1, #f~, if (((p=f[k,1])%2) == 1, p^4)); \\ Michel Marcus, Jul 11 2017

Scheme

(define (A005068 n) (cond ((= 1 n) 0) ((even? n) (A005068 (/ n 2))) (else (+ (A000583 (A020639 n)) (A005068 (A028234 n)))))) ;; Antti Karttunen, Jul 10 2017

Formula

Additive with a(p^e) = 0 if p = 2, p^4 otherwise.

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)^4 + a(A028234(n)).

a(n) = A005065(A000265(n)).

(End)

G.f.: Sum_{k>=2} prime(k)^4 * x^prime(k) / (1 - x^prime(k)). - Ilya Gutkovskiy, Aug 19 2021

Extensions

More terms from Antti Karttunen, Jul 10 2017