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A076577 Sum of squares of divisors d of n such that n/d is odd.

%I A076577 #35 Sep 15 2020 06:36:04
%S A076577 1,4,10,16,26,40,50,64,91,104,122,160,170,200,260,256,290,364,362,416,
%T A076577 500,488,530,640,651,680,820,800,842,1040,962,1024,1220,1160,1300,
%U A076577 1456,1370,1448,1700,1664,1682,2000,1850,1952,2366,2120,2210,2560,2451,2604
%N A076577 Sum of squares of divisors d of n such that n/d is odd.
%H A076577 Robert Israel, <a href="/A076577/b076577.txt">Table of n, a(n) for n = 1..10000</a>
%H A076577 J. W. L. Glaisher, <a href="https://books.google.com/books?id=bLs9AQAAMAAJ&amp;pg=RA1-PA1">On the representations of a number as the sum of two, four, six, eight, ten, and twelve squares</a>, Quart. J. Math. 38 (1907), 1-62 (see p. 4 and p. 8).
%H A076577 <a href="/index/Ge#Glaisher">Index entries for sequences mentioned by Glaisher</a>
%F A076577 G.f.: Sum_{m>0} m^2*x^m/(1-x^(2*m)). More generally, if b(n, k) is sum of k-th powers of divisors d of n such that n/d is odd then b(2n, k) = sigma_k(2n)-sigma_k(n), b(2n+1, k) = sigma_k(2n+1), where sigma_k(n) is sum of k-th powers of divisors of n. G.f. for b(n, k): Sum_{m>0} m^k*x^m/(1-x^(2*m)).
%F A076577 b(n, k) is multiplicative: b(2^e, k) = 2^(k*e), b(p^e, k) = (p^(ke+k)-1)/(p^k-1) for an odd prime p.
%F A076577 a(2*n) = sigma_2(2*n)-sigma_2(n), a(2*n+1) = sigma_2(2*n+1), where sigma_2(n) is sum of squares of divisors of n (cf. A001157).
%F A076577 b(n, k) = (sigma_k(2n)-sigma_k(n))/2^k. - _Vladeta Jovovic_, Oct 06 2003
%F A076577 Dirichlet g.f.: zeta(s)*(1-1/2^s)*zeta(s-2). - _Geoffrey Critzer_, Mar 28 2015
%F A076577 L.g.f.: -log(Product_{ k>0 } (x^k-1)^k/(x^k+1)^k)/2 = Sum_{ n>0 } (a(n)/n)*x^n. - _Benedict W. J. Irwin_, Jul 05 2016
%F A076577 Sum_{k=1..n} a(k) ~ 7*Zeta(3)*n^3 / 24. - _Vaclav Kotesovec_, Feb 08 2019
%e A076577 G.f. = x + 4*x^2 + 10*x^3 + 16*x^4 + 26*x^5 + 40*x^6 + 50*x^7 + 64*x^8 + ...
%p A076577 a:= n -> mul(`if`(t[1]=2, 2^(2*t[2]),
%p A076577      (t[1]^(2*(1+t[2]))-1)/(t[1]^2-1)),t=ifactors(n)[2]):
%p A076577 map(a, [$1..100]); # _Robert Israel_, Jul 05 2016
%t A076577 a[ n_] := If[ n < 1, 0, Sum[ d^2 Mod[ n/d, 2], {d, Divisors @ n}]]; (* _Michael Somos_, Jun 09 2014 *)
%t A076577 Table[CoefficientList[Series[-Log[Product[(x^k - 1)^k/(x^k + 1)^k, {k, 1, 80}]]/2, {x, 0, 80}], x][[n + 1]] n, {n, 1, 80}] (* _Benedict W. J. Irwin_, Jul 05 2016 *)
%t A076577 f[2, e_] := 4^e; f[p_, e_] := (p^(2*e + 2) - 1)/(p^2 - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* _Amiram Eldar_, Sep 15 2020 *)
%o A076577 (PARI) a(n) = sumdiv(n, d, d^2*((n/d) % 2)); \\ _Michel Marcus_, Jun 09 2014
%Y A076577 Cf. A001227, A002131, A001157, A050999.
%Y A076577 Glaisher's Delta'_i (i=0..12): A001227, A002131, A076577, A007331, A285989, A096960, A321817, A096961, A321818, A096962, A321819, A096963, A321820
%K A076577 mult,nonn
%O A076577 1,2
%A A076577 _Vladeta Jovovic_, Oct 19 2002