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

1,4

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.

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

Formula

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

Example

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

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}]

Crossrefs

Cf. A018818, A033630, A211110.

Sequence in context: A168177 A216864 A337175 * A286518 A029205 A229340

Adjacent sequences: A263429 A263430 A263431 * A263433 A263434 A263435

Keyword

nonn,look

Author

Geoffrey Critzer, Oct 18 2015

Status

approved