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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)
9
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-Fran├žois 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 A114319

Adjacent sequences:  A004098 A004099 A004100 * A004102 A004103 A004104

KEYWORD

nonn,nice,easy

AUTHOR

N. J. A. Sloane

EXTENSIONS

More terms, formula and better description from Christian G. Bower, Nov 15 1999

STATUS

approved

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Last modified January 18 03:13 EST 2019. Contains 319260 sequences. (Running on oeis4.)