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%I #17 Oct 28 2022 07:14:52
%S 1,10,17,53,37,170,65,236,174,370,145,901,197,650,629,987,325,1740,
%T 401,1961,1105,1450,577,4012,968,1970,1618,3445,901,6290,1025,4026,
%U 2465,3250,2405,9222,1445,4010,3349,8732,1765,11050,1937,7685,6438,5770
%N a(n) = Sum_{d|n} sigma(n*d).
%H Amiram Eldar, <a href="/A069546/b069546.txt">Table of n, a(n) for n = 1..10000</a>
%F Multiplicative with a(p^e) = (p^(e+1)*(p^(e+1)-1)-(p-1)*(e+1))/(p-1)^2.
%F Sum_{k=1..n} a(k) ~ c * n^3, where c = ((zeta(2)*zeta(3)^2)/3) * Product_{p prime} (1 + 1/p^2 - 1/p^4 - 1/p^5) = 1.09461730308... . - _Amiram Eldar_, Oct 28 2022
%t Table[ Apply[ Plus, DivisorSigma[1, n*Divisors[n]]], {n, 1, 50}]
%t f[p_, e_] := (p^(e + 1)*(p^(e + 1) - 1) - (p - 1)*(e + 1))/(p - 1)^2; a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* _Amiram Eldar_, Oct 28 2022 *)
%o (Magma) [&+[DivisorSigma(1,n*d):d in Divisors(n)]:n in [1..50]]; // _Marius A. Burtea_, Sep 15 2019
%o (PARI) a(n) = sumdiv(n, d, sigma(n*d)); \\ _Michel Marcus_, Sep 15 2019
%Y Cf. A061391, A062354.
%Y Cf. A002117, A013661.
%K mult,easy,nonn
%O 1,2
%A _Vladeta Jovovic_, Apr 17 2002
%E Edited and extended by _Robert G. Wilson v_, Apr 22 2002
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