A004101 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. (Formerly M0753)

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

Offset

0,3

Comments

The number of semisimple rings with p^n elements does not depend on the prime number p. - Paul Laubie, Mar 05 2024

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

N. J. A. Sloane

Extensions

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

Name clarified by Paul Laubie, Mar 05 2024

Status

approved