cp's OEIS Frontend

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

%I A005068 #22 Jun 20 2022 04:21:48
%S A005068 0,0,81,0,625,81,2401,0,81,625,14641,81,28561,2401,706,0,83521,81,
%T A005068 130321,625,2482,14641,279841,81,625,28561,81,2401,707281,706,923521,
%U A005068 0,14722,83521,3026,81,1874161,130321,28642,625,2825761,2482,3418801,14641,706,279841,4879681,81,2401,625,83602,28561,7890481,81
%N A005068 Sum of 4th powers of odd primes dividing n.
%H A005068 Antti Karttunen, <a href="/A005068/b005068.txt">Table of n, a(n) for n = 1..10000</a>
%F A005068 Additive with a(p^e) = 0 if p = 2, p^4 otherwise.
%F A005068 From _Antti Karttunen_, Jul 10 2017: (Start)
%F A005068 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 A005068 a(n) = A005065(A000265(n)).
%F A005068 (End)
%F A005068 G.f.: Sum_{k>=2} prime(k)^4 * x^prime(k) / (1 - x^prime(k)). - _Ilya Gutkovskiy_, Aug 19 2021
%t A005068 Array[DivisorSum[#, #^4 &, And[PrimeQ@ #, OddQ@ #] &] &, 54] (* _Michael De Vlieger_, Jul 11 2017 *)
%t A005068 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 A005068 (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 A005068 (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 A005068 Cf. A000265, A000583, A005065, A005066, A005067, A005069, A020639, A028234.
%K A005068 nonn
%O A005068 1,3
%A A005068 _N. J. A. Sloane_
%E A005068 More terms from _Antti Karttunen_, Jul 10 2017