A300977 Number of partitions of n into parts having the same number of divisors as n.

1, 1, 1, 1, 1, 2, 1, 3, 1, 1, 1, 6, 1, 9, 2, 1, 1, 17, 1, 23, 1, 2, 4, 40, 1, 2, 5, 3, 1, 87, 1, 111, 2, 5, 10, 7, 1, 219, 14, 9, 1, 336, 1, 413, 3, 1, 26, 614, 1, 4, 3, 30, 4, 1083, 2, 42, 1, 51, 66, 1850, 1, 2198, 91, 2, 1, 95, 2, 3630, 9, 130, 2, 5007, 1, 5861, 209, 3, 11, 232, 3, 9282, 1

Offset

0,6

Links

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Index entries for sequences related to partitions

Formula

a(n) = [x^n] Product_{d(k) = d(n)} 1/(1 - x^k).

Example

a(27) = 3 because we have [27], [21, 6] and [15, 6, 6], where 27, 21, 15 and 6 are numbers with 4 divisors.

Maple

with(numtheory):
a:= proc(m) option remember; local k, b; k, b:= tau(m),
      proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
        b(n, i-1)+`if`(tau(i)=k, b(n-i, min(i, n-i)), 0)))
      end: b(m$2)
    end:
seq(a(n), n=0..100);  # Alois P. Heinz, Mar 17 2018

Mathematica

Table[SeriesCoefficient[Product[1/(1 - Boole[DivisorSigma[0, k] == DivisorSigma[0, n]] x^k), {k, 1, n}], {x, 0, n}], {n, 0, 80}]

Crossrefs

Cf. A000005, A300978, A300979, A300980, A300982, A300983.

Sequence in context: A057021 A152443 A119804 * A144869 A247564 A193870

Adjacent sequences: A300974 A300975 A300976 * A300978 A300979 A300980

Keyword

nonn

Author

Ilya Gutkovskiy, Mar 17 2018

Status

approved