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A009490
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Number of distinct orders of permutations of n objects; number of nonisomorphic cyclic subgroups of symmetric group S_n.
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15
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1, 1, 2, 3, 4, 6, 6, 9, 11, 14, 16, 20, 23, 27, 31, 35, 43, 47, 55, 61, 70, 78, 88, 98, 111, 123, 136, 152, 168, 187, 204, 225, 248, 271, 296, 325, 356, 387, 418, 455, 495, 537, 581, 629, 678, 732, 787, 851, 918, 986, 1056, 1133, 1217, 1307, 1399, 1498, 1600, 1708, 1823
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OFFSET
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0,3
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COMMENTS
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Also number of different LCM's of partitions of n.
a(n) <= A023893(n), which counts the nonisomorphic Abelian subgroups of S_n. - M. F. Hasler, May 24 2013
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LINKS
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FORMULA
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a(n) = Sum_{k=0..n} b(k), where b(k) is the number of partitions of k into distinct prime power parts (1 excluded) (A051613). - Vladeta Jovovic
G.f.: Prod(p prime, 1 + Sum(k >= 1, x^(p^k))) / (1-x). - David W. Wilson, Apr 19, 2000
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MAPLE
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b:= proc(n, i) option remember; local p;
p:= `if`(i<1, 1, ithprime(i));
`if`(n=0 or i<1, 1, b(n, i-1)+
add(b(n-p^j, i-1), j=1..ilog[p](n)))
end:
a:= n-> b(n, numtheory[pi](n)):
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MATHEMATICA
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Table[ Length[ Union[ Apply[ LCM, Partitions[ n ], 1 ] ] ], {n, 30} ]
f[n_] := Length@ Union[LCM @@@ IntegerPartitions@ n]; Array[f, 60, 0]
(* Caution, the following is Extremely Slow and Resource Intensive *) CoefficientList[ Series[ Expand[ Product[1 + Sum[x^(Prime@ i^k), {k, 4}], {i, 10}]/(1 - x)], {x, 0, 30}], x]
b[n_, i_] := b[n, i] = Module[{p}, p = If[i<1, 1, Prime[i]]; If[n == 0 || i<1, 1, b[n, i-1]+Sum[b[n-p^j, i-1], {j, 1, Log[p, n]}]]]; a[n_] := b[n, PrimePi[n]]; Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Feb 03 2014, after Alois P. Heinz *)
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PROG
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(PARI) /* compute David W. Wilson's g.f., needs <1 sec for 1000 terms */
N=1000; x='x+O('x^N); /* N terms */
gf=1; /* generating function */
{ forprime(p=2, N,
sm = 1; pp=p; /* sum; prime power */
while ( pp<N, sm += x^pp; pp *= p; );
gf *= sm; /* update g.f. */
); }
gf/=(1-x); /* cumulative sums */
Vec(gf) /* show terms */ /* Joerg Arndt, Jan 19 2011 */
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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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STATUS
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approved
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