A339409 Number of compositions (ordered partitions) of n into an odd number of primes.

0, 0, 1, 1, 0, 1, 1, 4, 3, 4, 7, 12, 19, 22, 32, 53, 80, 120, 160, 245, 368, 553, 800, 1164, 1736, 2588, 3813, 5598, 8226, 12228, 18060, 26657, 39221, 57945, 85656, 126506, 186584, 275307, 406514, 600488, 886255, 1308088, 1930648, 2850861, 4208743, 6212824, 9170440, 13538025

Offset

0,8

Links

Table of n, a(n) for n=0..47.

Index entries for sequences related to compositions

Formula

G.f.: (1/2) * (1 / (1 - Sum_{k>=1} x^prime(k)) - 1 / (1 + Sum_{k>=1} x^prime(k))).

Example

a(8) = 3 because we have [3, 3, 2], [3, 2, 3] and [2, 3, 3].

Maple

b:= proc(n, t) option remember; `if`(n=0, t, add(
      b(n-ithprime(j), 1-t), j=1..numtheory[pi](n)))
    end:
a:= n-> b(n, 0):
seq(a(n), n=0..55);  # Alois P. Heinz, Dec 03 2020

Mathematica

nmax = 47; CoefficientList[Series[(1/2) (1/(1 - Sum[x^Prime[k], {k, 1, nmax}]) - 1/(1 + Sum[x^Prime[k], {k, 1, nmax}])), {x, 0, nmax}], x]

Crossrefs

Cf. A000040, A023360, A166444, A184199, A339381, A339408.

Sequence in context: A289672 A359864 A361849 * A075246 A257840 A132984

Adjacent sequences: A339406 A339407 A339408 * A339410 A339411 A339412

Keyword

nonn

Author

Ilya Gutkovskiy, Dec 03 2020

Status

approved