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%I #26 Nov 05 2022 08:18:37
%S 1,63,364,2047,3906,22932,19608,65535,88573,246078,177156,745108,
%T 402234,1235304,1421784,2097151,1508598,5580099,2613660,7995582,
%U 7137312,11160828,6728904,23854740,12207031,25340742,21523360,40137576,21243690,89572392,29583456,67108863
%N a(n) = sigma(n^5).
%C a(n) modulo 6 begins: [1,3,4,1,0,0,0,3,1,0,0,4,0,0,0,1,0,3,0,0,0,0,0,0,1,0,...], in which positions of nonzero residues seem related to squares.
%H Seiichi Manyama, <a href="/A203556/b203556.txt">Table of n, a(n) for n = 1..10000</a>
%F Logarithmic derivative of A203557.
%F Multiplicative with a(p^e) = (p^(5*e+1)-1)/(p-1) for prime p. - _Andrew Howroyd_, Jul 23 2018
%F From _Amiram Eldar_, Nov 05 2022: (Start)
%F a(n) = A000203(A000584(n)) = A000203(n^5).
%F Sum_{k=1..n} a(k) ~ c * n^6, where c = (zeta(6)/6) * Product_{p prime} (1 + 1/p^2 + 1/p^3 + 1/p^4 + 1/p^5) = 0.3220880186... . (End)
%e L.g.f.: L(x) = x + 63/2*x^2 + 364/3*x^3 + 2047/4*x^4 + 3906/5*x^5 +...
%e where the g.f. of A203557 begins:
%e exp(L(x)) = 1 + x + 32*x^2 + 153*x^3 + 1145*x^4 + 5677*x^5 + 37641*x^6 +...
%t f[p_, e_] := (p^(5*e + 1) - 1)/(p - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 50] (* _Amiram Eldar_, Sep 09 2020 *)
%t DivisorSigma[1,Range[40]^5] (* _Harvey P. Dale_, Dec 05 2021 *)
%o (PARI) a(n) = sigma(n^5)
%Y Cf. A203557 (exp), A000203 (sigma), A000584, A013664.
%Y Variants: A065764, A175926, A202994.
%K nonn,easy,mult
%O 1,2
%A _Paul D. Hanna_, Jan 03 2012
%E Keyword:mult added by _Andrew Howroyd_, Jul 23 2018
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