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%I #20 Jul 30 2017 22:52:41
%S 1,1,2,3,4,5,7,12,15,22,28,38,45,52,81,107,130,179,194,280,348,438,
%T 502,693,848,1037,1274,1594,1847,2473,2851,3652,4271,5137,6140,7995,
%U 9103,11046,12978,16216,18348,23153,26239,31880,37582,45144,51469,63571,71910
%N a(n) is the number of different degrees in the sequence of the degrees of the irreducible representations of the symmetric group S_n, i.e., count each degree only once.
%C The total number of irreducible representations of S_n is the partition function p(n) (sequence A000041) - this is the total number of the degrees counting multiplicities.
%C Also a(n) = number of distinct values of A153452(m) when A056239(m) is equal to n. - _Naohiro Nomoto_, Dec 31 2008
%e a(6) = 5 because the degrees for S_6 are 1,1,5,5,5,5,9,9,10,10,16 and counting each degree only once only 5 numbers remain: 1,5,9,10,16.
%p with(numtheory):
%p g:= proc(n) option remember; `if`(n=1, 1,
%p add(g(n/q*`if`(q=2, 1, prevprime(q))), q=factorset(n)))
%p end:
%p b:= proc(n, i) option remember; `if`(n=0 or i<2, [2^n],
%p [seq(map(p->p*ithprime(i)^j, b(n-i*j, i-1))[], j=0..n/i)])
%p end:
%p a:= n-> nops(map(g, {b(n, n)[]})):
%p seq(a(n), n=1..30); # _Alois P. Heinz_, Aug 09 2012
%t g[n_] := g[n] = If[n == 1, 1, Sum[g[n/q*If[q == 2, 1, NextPrime[q, -1]]], {q, FactorInteger[n][[All, 1]]}]]; b[n_, i_] :=b[n, i] = If[n == 0 || i<2, {2^n}, Flatten @ Table[ Map[Function[{p}, p*Prime[i]^j], b[n-i*j, i-1]], {j, 0, n/i}] ]; a[n_] := Length[Union[g /@ b[n, n]]]; Table[a[n], {n, 1, 30}] (* _Jean-François Alcover_, Apr 15 2015, after _Alois P. Heinz_ *)
%Y Cf. A000041, A060240, A215366.
%K nonn
%O 1,3
%A Avi Peretz (njk(AT)netvision.net.il), Apr 07 2001
%E More terms from _Vladeta Jovovic_, May 20 2003
%E a(22)-a(49) from _Alois P. Heinz_, Aug 09 2012
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