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Sum of 4th powers of odd primes dividing n.
7

%I #22 Jun 20 2022 04:21:48

%S 0,0,81,0,625,81,2401,0,81,625,14641,81,28561,2401,706,0,83521,81,

%T 130321,625,2482,14641,279841,81,625,28561,81,2401,707281,706,923521,

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

%N Sum of 4th powers of odd primes dividing n.

%H Antti Karttunen, <a href="/A005068/b005068.txt">Table of n, a(n) for n = 1..10000</a>

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

%F From _Antti Karttunen_, Jul 10 2017: (Start)

%F a(1) = 0; after which, for even n: a(n) = a(n/2), for odd n: a(n) = A020639(n)^4 + a(A028234(n)).

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

%F (End)

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

%t Array[DivisorSum[#, #^4 &, And[PrimeQ@ #, OddQ@ #] &] &, 54] (* _Michael De Vlieger_, Jul 11 2017 *)

%t 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 *)

%o (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

%o (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

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

%K nonn

%O 1,3

%A _N. J. A. Sloane_

%E More terms from _Antti Karttunen_, Jul 10 2017