A298604 -id:A298604 - cp's OEIS frontend

A298603 Number of partitions of n into odd prime parts (including 1).

1, 1, 1, 2, 2, 3, 4, 5, 6, 7, 9, 11, 13, 16, 19, 22, 26, 31, 36, 42, 49, 56, 65, 75, 86, 98, 112, 127, 144, 164, 185, 209, 235, 264, 297, 332, 372, 416, 463, 516, 574, 638, 708, 785, 869, 960, 1061, 1171, 1291, 1421, 1563, 1718, 1886, 2070, 2269, 2484, 2718, 2972, 3247, 3545, 3868, 4216, 4592

Offset: 0

Ilya Gutkovskiy, Jan 22 2018

nonn

Partial sums of A099773.

			a(6) = 4 because we have [5, 1], [3, 3], [3, 1, 1, 1] and [1, 1, 1, 1, 1, 1].

Index entries for related partition-counting sequences

Cf. A000607, A006005, A008578, A034891, A099773, A298604.

nmax = 62; CoefficientList[Series[1/(1 - x) Product[1/(1 - x^Prime[k]), {k, 2, nmax}], {x, 0, nmax}], x]

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

A309676 Number of compositions (ordered partitions) of n into odd primes (including 1).

Original entry on oeis.org

1, 1, 1, 2, 3, 5, 8, 13, 21, 33, 53, 86, 138, 222, 357, 573, 921, 1481, 2381, 3828, 6153, 9890, 15898, 25556, 41082, 66039, 106156, 170644, 274307, 440945, 708815, 1139412, 1831589, 2944253, 4732847, 7607989, 12229743, 19659153, 31601828, 50799517, 81659549

Offset: 0

Ilya Gutkovskiy, Aug 12 2019

nonn

Index entries for sequences related to compositions

Cf. A002124, A006005, A008578, A023360, A080545, A280917, A298603, A298604.

a:= proc(n) option remember; `if`(n=0, 1, a(n-1)+
      add(`if`(isprime(j), a(n-j), 0), j=3..n))
    end:
seq(a(n), n=0..42);  # Alois P. Heinz, Aug 12 2019

nmax = 42; CoefficientList[Series[1/(1 - x - Sum[x^Prime[k], {k, 2, nmax}]), {x, 0, nmax}], x]
a[0] = 1; a[n_] := a[n] = Sum[Boole[PrimeOmega[k] < 2 && OddQ[k]] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 42}]

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

cp's OEIS Frontend

A298603 Number of partitions of n into odd prime parts (including 1).

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