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%I #14 May 20 2024 02:28:51
%S 1,32,243,1054,3125,7776,16807,33818,59289,100000,161051,256122,
%T 371293,537824,759375,1082386,1419857,1897248,2476099,3293750,4084101,
%U 5153632,6436343,8217774,9768745,11881376,14408187,17714578,20511149,24300000,28629151,34636802,39135393
%N a(n) = Sum_{k=1..n} sigma( (n/gcd(k,n))^4 ).
%H Amiram Eldar, <a href="/A373000/b373000.txt">Table of n, a(n) for n = 1..10000</a>
%F If k is squarefree (cf. A005117) then a(k) = k^5.
%F a(n) = Sum_{d|n} phi(d) * sigma(d^4).
%F From _Amiram Eldar_, May 20 2024: (Start)
%F Multiplicative with a(p^e) = (p^(5*e+5)-1)/(p^5-1) - (p^e-1)/(p-1).
%F Sum_{k=1..n} a(k) ~ c * n^6 / 6, where c = (zeta(5) * zeta(6) / zeta(2)) * Product_{p prime} (1 + 1/p^2 + 1/p^4 - 1/p^5) = 1.00815225456201682259... . (End)
%t a[n_] := DivisorSum[n, EulerPhi[#] * DivisorSigma[1, #^4] &]; Array[a, 100] (* _Amiram Eldar_, May 20 2024 *)
%o (PARI) a(n) = sumdiv(n, d, eulerphi(d)*sigma(d^4));
%Y Cf. A005117.
%Y Cf. A013663, A013664.
%K nonn,mult
%O 1,2
%A _Seiichi Manyama_, May 19 2024
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