A060437 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.

1, 1, 2, 3, 4, 5, 7, 12, 15, 22, 28, 38, 45, 52, 81, 107, 130, 179, 194, 280, 348, 438, 502, 693, 848, 1037, 1274, 1594, 1847, 2473, 2851, 3652, 4271, 5137, 6140, 7995, 9103, 11046, 12978, 16216, 18348, 23153, 26239, 31880, 37582, 45144, 51469, 63571, 71910

Offset: 1

Author

Avi Peretz (njk(AT)netvision.net.il), Apr 07 2001

Keywords

• nonn

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

Examples

			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.
		

Crossrefs

Cf. A000041, A060240, A215366.

Programs

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 *)

Extensions

More terms from Vladeta Jovovic, May 20 2003

a(22)-a(49) from Alois P. Heinz, Aug 09 2012