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A004101
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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)
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15
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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
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OFFSET
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0,3
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COMMENTS
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The number of semisimple rings with p^n elements does not depend on the prime number p. - Paul Laubie, Mar 05 2024
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REFERENCES
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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).
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LINKS
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FORMULA
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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
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EXAMPLE
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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.
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MAPLE
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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:
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
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MATHEMATICA
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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 *)
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PROG
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(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 */
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CROSSREFS
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KEYWORD
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nonn,nice,easy
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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