A357311 Number of partitions of n into divisors of n that are smaller than sqrt(n).

1, 0, 1, 1, 1, 1, 4, 1, 5, 1, 6, 1, 19, 1, 8, 6, 9, 1, 37, 1, 36, 8, 12, 1, 169, 1, 14, 10, 64, 1, 247, 1, 81, 12, 18, 8, 478, 1, 20, 14, 405, 1, 512, 1, 144, 82, 24, 1, 2825, 1, 146, 18, 196, 1, 1000, 12, 743, 20, 30, 1, 19858, 1, 32, 112, 289, 14, 1728, 1, 324, 24, 1105

Offset

0,7

Links

Alois P. Heinz, Table of n, a(n) for n = 0..10000

Formula

a(n) = [x^n] Product_{d|n, d < sqrt(n)} 1 / (1 - x^d).

Maple

a:= proc(n) option remember; uses numtheory; local b, l;
      l:= sort([select(x-> is(x<sqrt(n)), divisors(n))[]]):
      b:= proc(m, i) option remember; `if`(m=0 or i=1, 1,
            `if`(i<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=0..70);  # Alois P. Heinz, Sep 23 2022

Mathematica

a[n_] := SeriesCoefficient[Product[1/(1 - Boole[d < Sqrt[n]] x^d), {d, Divisors[n]}], {x, 0, n}]; Table[a[n], {n, 0, 70}]

Crossrefs

Cf. A018818, A210442, A293813, A327642, A357312.

Sequence in context: A323413 A019303 A339967 * A373364 A107463 A157104

Adjacent sequences: A357308 A357309 A357310 * A357312 A357313 A357314

Keyword

nonn

Author

Ilya Gutkovskiy, Sep 23 2022

Status

approved