OFFSET
1,3
COMMENTS
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.
Also a(n) = number of distinct values of A153452(m) when A056239(m) is equal to n. - Naohiro Nomoto, Dec 31 2008
EXAMPLE
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.
MAPLE
with(numtheory):
g:= proc(n) option remember; `if`(n=1, 1,
add(g(n/q*`if`(q=2, 1, prevprime(q))), q=factorset(n)))
end:
b:= proc(n, i) option remember; `if`(n=0 or i<2, [2^n],
[seq(map(p->p*ithprime(i)^j, b(n-i*j, i-1))[], j=0..n/i)])
end:
a:= n-> nops(map(g, {b(n, n)[]})):
seq(a(n), n=1..30); # Alois P. Heinz, Aug 09 2012
MATHEMATICA
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 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Avi Peretz (njk(AT)netvision.net.il), Apr 07 2001
EXTENSIONS
More terms from Vladeta Jovovic, May 20 2003
a(22)-a(49) from Alois P. Heinz, Aug 09 2012
STATUS
approved