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A279364 Sum of 5th powers of proper divisors of n.

%I A279364 #30 Feb 16 2025 08:33:37
%S A279364 0,1,1,33,1,276,1,1057,244,3158,1,9076,1,16840,3369,33825,1,67101,1,
%T A279364 104182,17051,161084,1,290676,3126,371326,59293,555688,1,870552,1,
%U A279364 1082401,161295,1419890,19933,2206525,1,2476132,371537,3336950,1,4646784,1,5315740,821793,6436376,1,9301876,16808,9868783
%N A279364 Sum of 5th powers of proper divisors of n.
%H A279364 Amiram Eldar, <a href="/A279364/b279364.txt">Table of n, a(n) for n = 1..10000</a>
%H A279364 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/ProperDivisor.html">Proper divisors</a>.
%H A279364 <a href="/index/Su#sums_of_divisors">Index entries for sequences related to sums of divisors</a>.
%F A279364 a(n) = 1 if n is prime.
%F A279364 a(p^k) = (p^(5*k) - 1)/(p^5 - 1) when p is prime.
%F A279364 Dirichlet g.f.: zeta(s-5)*(zeta(s) - 1).
%F A279364 a(n) = A001160(n) - A000584(n).
%F A279364 G.f.: -x*(1 + 26*x + 66*x^2 + 26*x^3 + x^4)/(1 - x)^6 + Sum_{k>=1} k^5 x^k/(1 - x^k). - _Ilya Gutkovskiy_, Mar 18 2017
%F A279364 Sum_{k=1..n} a(k) ~ (zeta(6) - 1) * n^6 / 6. - _Amiram Eldar_, Jan 11 2025
%e A279364 a(10) = 1^5 + 2^5 + 5^5 = 3158, because 10 has 3 proper divisors {1,2,5}.
%e A279364 a(11) = 1^5 = 1, because 11 has 1 proper divisor {1}.
%t A279364 Table[DivisorSigma[5, n] - n^5, {n, 50}]
%o A279364 (PARI) for(n=1, 50, print1(sigma(n, 5) - n^5,", ")) \\ _Indranil Ghosh_, Mar 18 2017
%o A279364 (Python)
%o A279364 from sympy.ntheory import divisor_sigma
%o A279364 print([divisor_sigma(n,5) - n**5 for n in range(1,51)]) # _Indranil Ghosh_, Mar 18 2017
%Y A279364 Cf. A000584, A001160, A013664.
%Y A279364 Cf. A001065, A067558, A276634, A279363.
%K A279364 nonn,easy
%O A279364 1,4
%A A279364 _Ilya Gutkovskiy_, Dec 10 2016