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%I M0753 #50 Mar 06 2024 04:52:43
%S 1,1,2,3,6,8,13,18,29,40,58,79,115,154,213,284,391,514,690,900,1197,
%T 1549,2025,2600,3377,4306,5523,7000,8922,11235,14196,17777,22336,
%U 27825,34720,43037,53446,65942,81423,100033,122991,150481,184149,224449,273614,332291
%N Number of partitions of n of the form a_1*b_1^2 + a_2*b_2^2 + ...; number of semisimple rings with p^n elements for any prime p.
%C The number of semisimple rings with p^n elements does not depend on the prime number p. - _Paul Laubie_, Mar 05 2024
%D J. Knopfmacher, Abstract Analytic Number Theory. North-Holland, Amsterdam, 1975, p. 293.
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Alois P. Heinz and Vaclav Kotesovec, <a href="/A004101/b004101.txt">Table of n, a(n) for n = 0..10000</a> (terms 0..1000 from Alois P. Heinz)
%H Gert Almkvist, <a href="https://arxiv.org/abs/math/0612446">Asymptotics of various partitions</a>, arXiv:math/0612446 [math.NT], 2006 (section 6).
%H I. G. Connell, <a href="http://cms.math.ca/10.4153/CMB-1964-002-1">A number theory problem concerning finite groups and rings</a>, Canad. Math. Bull, 7 (1964), 23-34.
%H I. G. Connell, <a href="/A000688/a000688.pdf">Letter to N. J. A. Sloane, no date</a>
%H N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%F EULER transform of A046951.
%F a(n) ~ exp(Pi^2 * sqrt(n) / 3 + sqrt(3/(2*Pi)) * Zeta(1/2) * Zeta(3/2) * n^(1/4) - 9 * Zeta(1/2)^2 * Zeta(3/2)^2 / (16*Pi^3)) * Pi^(3/4) / (sqrt(2) * 3^(1/4) * n^(5/8)) [Almkvist, 2006]. - _Vaclav Kotesovec_, Jan 03 2017
%e 4 = 4*1^2 = 1*2^2 = 3*1^2 + 1*1^2 = 2*1^2 + 2*1^2 = 2*1^2 + 1*1^2 + 1*1^2 = 1*1^2 + 1*1^2 + 1*1^2 + 1*1^2.
%p with(numtheory):
%p a:= proc(n) option remember;
%p `if`(n=0, 1, add(add(d* mul(1+iquo(i[2], 2),
%p i=ifactors(d)[2]), d=divisors(j))*a(n-j), j=1..n)/n)
%p end:
%p seq(a(n), n=0..60); # _Alois P. Heinz_, Nov 26 2013
%p sqd:=proc(n) local t1,d; t1:=0; for d in divisors(n) do if (n mod d^2) = 0 then t1:=t1+1; fi; od; t1; end; # A046951
%p t2:=mul( 1/(1-x^n)^sqd(n),n=1..65); series(t2,x,60); seriestolist(%); # _N. J. A. Sloane_, Jun 24 2015
%t max = 45; A046951 = Table[Sum[Floor[n/k^2], {k, n}], {n, 0, max}] // Differences; f = Product[1/(1-x^n)^A046951[[n]], {n, 1, max}]; CoefficientList[Series[f, {x, 0, max}], x] (* _Jean-François Alcover_, Feb 11 2014 *)
%t nmax = 50; CoefficientList[Series[Product[1/(1 - x^(j*k^2)), {k, 1, Floor[Sqrt[nmax]] + 1}, {j, 1, Floor[nmax/k^2] + 1}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Jan 03 2017 *)
%o (PARI) N=66; x='x+O('x^N); gf=1/prod(j=1,N, eta(x^(j^2))); Vec(gf) /* _Joerg Arndt_, May 03 2008 */
%Y Cf. A006171, A038538, A280451, A280661, A280662.
%K nonn,nice,easy
%O 0,3
%A _N. J. A. Sloane_
%E More terms, formula and better description from _Christian G. Bower_, Nov 15 1999
%E Name clarified by _Paul Laubie_, Mar 05 2024
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