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A074761
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Number of partitions of n of order n.
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37
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1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 1, 9, 1, 4, 5, 1, 1, 12, 1, 27, 7, 6, 1, 81, 1, 7, 1, 54, 1, 407, 1, 1, 11, 9, 13, 494, 1, 10, 13, 423, 1, 981, 1, 137, 115, 12, 1, 1309, 1, 59, 17, 193, 1, 240, 21, 1207, 19, 15, 1, 47274, 1, 16, 239, 1, 25, 3284, 1, 333, 23, 3731, 1, 42109, 1, 19
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OFFSET
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1,6
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COMMENTS
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Order of partition is lcm of its parts.
a(n) is the number of conjugacy classes of the symmetric group S_n such that a representative of the class has order n. Here order means the order of an element of a group. Note that a(n) = 1 if and only if n is a prime power. - W. Edwin Clark, Aug 05 2014
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LINKS
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FORMULA
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Coefficient of x^n in expansion of Sum_{i divides n} A008683(n/i)*1/Product_{j divides i} (1-x^j).
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EXAMPLE
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The a(15) = 5 partitions are (15), (5,3,3,3,1), (5,5,3,1,1), (5,3,3,1,1,1,1), (5,3,1,1,1,1,1,1,1). - Gus Wiseman, Aug 01 2018
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MAPLE
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A:= proc(n)
uses numtheory;
local S;
S:= add(mobius(n/i)*1/mul(1-x^j, j=divisors(i)), i=divisors(n));
coeff(series(S, x, n+1), x, n);
end proc:
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MATHEMATICA
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a[n_] := With[{s = Sum[MoebiusMu[n/i]*1/Product[1-x^j, {j, Divisors[i]}], {i, Divisors[n]}]}, SeriesCoefficient[s, {x, 0, n}]]; Array[a, 80}] (* Jean-François Alcover, Feb 29 2016 *)
Table[Length[Select[IntegerPartitions[n], LCM@@#==n&]], {n, 50}] (* Gus Wiseman, Aug 01 2018 *)
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PROG
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(PARI)
pr(k, x)={my(t=1); fordiv(k, d, t *= (1-x^d) ); return(t); }
a(n) =
{
my( x = 'x+O('x^(n+1)) );
polcoeff( Pol( sumdiv(n, i, moebius(n/i) / pr(i, x) ) ), n );
}
vector(66, n, a(n) )
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CROSSREFS
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KEYWORD
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easy,nonn
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AUTHOR
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STATUS
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approved
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