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
KEYWORD
nonn,nice,easy
AUTHOR
EXTENSIONS
More terms, formula and better description from Christian G. Bower, Nov 15 1999
Name clarified by Paul Laubie, Mar 05 2024
STATUS
approved