A263432 Number of partitions of n into divisors of n with at most one 1.

1, 1, 1, 2, 1, 4, 1, 4, 2, 4, 1, 17, 1, 4, 4, 10, 1, 24, 1, 24, 4, 4, 1, 126, 2, 4, 5, 30, 1, 171, 1, 36, 4, 4, 4, 490, 1, 4, 4, 251, 1, 290, 1, 43, 42, 4, 1, 1822, 2, 50, 4, 50, 1, 462, 4, 421, 4, 4, 1, 13284, 1, 4, 49, 202, 4, 616, 1, 63, 4, 581

Offset: 1

Author

Geoffrey Critzer, Oct 18 2015

Keywords

• nonn
• look

Comments

a(n) is also the number of ways to partition a group of order n into its center and its nontrivial conjugacy classes. That is, the number of possible sums in the class equation.

Examples

			a(15) = 4 because we have: [15], [5,5,5], [5,3,3,3,1], [3,3,3,3,3].
		

References

• D. S. Dummit and R. M. Foote, Abstract Algebra, Wiley, 3rd edition 2003, page 124.

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A018818, A033630, A211110.

Programs

Maple

with(numtheory):
a:= proc(n) local b, l; l:= sort([(divisors(n) minus {1})[]]):
      b:= proc(m, i) option remember; `if`(m=0, 1, `if`(i<1,
           `if`(m=1, 1, 0), b(m, i-1)+`if`(l[i]>m, 0, b(m-l[i], i))))
          end; forget(b):
      b(n, nops(l))
    end:
seq(a(n), n=1..100);  # Alois P. Heinz, Oct 18 2015

Mathematica

Table[d = Drop[Divisors[n], 1];Coefficient[Series[(1 + x)/Product[1 - x^d[[i]], {i, Length[d]}], {x, 0, n}], x,n], {n, 70}]

Formula

a(n) is the coefficient of x^n in the expansion of (1 + x)/Product_{d>1,d divides n} (1 - x^d).