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 A004101 Number of partitions of n of the form a_1*b_1^2 + a_2*b_2^2 + ...; number of semi-simple rings with n elements. (Formerly M0753) 12
 1, 1, 2, 3, 6, 8, 13, 18, 29, 40, 58, 79, 115, 154, 213, 284, 391, 514, 690, 900, 1197, 1549, 2025, 2600, 3377, 4306, 5523, 7000, 8922, 11235, 14196, 17777, 22336, 27825, 34720, 43037, 53446, 65942, 81423, 100033, 122991, 150481, 184149, 224449, 273614, 332291 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 REFERENCES J. Knopfmacher, Abstract Analytic Number Theory. North-Holland, Amsterdam, 1975, p. 293. N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence). LINKS Alois P. Heinz and Vaclav Kotesovec, Table of n, a(n) for n = 0..10000 (terms 0..1000 from Alois P. Heinz) Gert Almkvist, Asymptotics of various partitions, arXiv:math/0612446 [math.NT], 2006 (section 6). I. G. Connell, A number theory problem concerning finite groups and rings, Canad. Math. Bull, 7 (1964), 23-34. I. G. Connell, Letter to N. J. A. Sloane, no date N. J. A. Sloane, Transforms FORMULA EULER transform of A046951. 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 EXAMPLE 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. MAPLE with(numtheory): a:= proc(n) option remember;       `if`(n=0, 1, add(add(d* mul(1+iquo(i[2], 2),       i=ifactors(d)[2]), d=divisors(j))*a(n-j), j=1..n)/n)     end: seq(a(n), n=0..60);  # Alois P. Heinz, Nov 26 2013 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 t2:=mul( 1/(1-x^n)^sqd(n), n=1..65); series(t2, x, 60); seriestolist(%); # N. J. A. Sloane, Jun 24 2015 MATHEMATICA 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 *) 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 *) PROG (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 */ CROSSREFS Cf. A006171, A038538, A280451, A280661, A280662. Sequence in context: A024788 A285472 A318027 * A003405 A153918 A308909 Adjacent sequences:  A004098 A004099 A004100 * A004102 A004103 A004104 KEYWORD nonn,nice,easy AUTHOR EXTENSIONS More terms, formula and better description from Christian G. Bower, Nov 15 1999 STATUS approved

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Last modified June 2 14:21 EDT 2020. Contains 334787 sequences. (Running on oeis4.)