A286225 -id:A286225 - cp's OEIS frontend

A286224 Number of compositions (ordered partitions) of n into parts with an odd number of distinct prime divisors.

1, 0, 1, 1, 2, 3, 4, 8, 11, 19, 28, 47, 72, 116, 182, 289, 460, 724, 1153, 1820, 2891, 4572, 7249, 11482, 18190, 28821, 45651, 72338, 114582, 181549, 287597, 455647, 721849, 1143590, 1811753, 2870247, 4547245, 7203933, 11412922, 18080907, 28644799, 45380602, 71894401, 113899027, 180444897, 285870668

Offset: 0

Ilya Gutkovskiy, May 04 2017

nonn

			a(6) = 4 because we have [4, 2], [3, 3], [2, 4] and [2, 2, 2].

Amiram Eldar, Table of n, a(n) for n = 0..5000
Eric Weisstein's World of Mathematics, Distinct Prime Factors
Index entries for sequences related to compositions

Cf. A030230, A285799, A286220, A286225.

nmax = 45; CoefficientList[Series[1/(1 - Sum[Boole[OddQ[PrimeNu[k]]] x^k, {k, 1, nmax}]), {x, 0, nmax}], x]

G.f.: 1/(1 - Sum_{k>=1} x^A030230(k)).

A286227 Number of compositions (ordered partitions) of n into parts with an even number of prime divisors (counted with multiplicity).

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 5, 7, 10, 15, 24, 36, 53, 78, 118, 179, 271, 405, 605, 907, 1366, 2055, 3086, 4628, 6948, 10440, 15689, 23560, 35371, 53110, 79771, 119821, 179958, 270243, 405833, 609495, 915394, 1374780, 2064647, 3100680, 4656676, 6993575, 10503180, 15773877, 23689467, 35577360

Offset: 0

Ilya Gutkovskiy, May 04 2017

nonn

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

Amiram Eldar, Table of n, a(n) for n = 0..5000
Eric Weisstein's World of Mathematics, Prime Factor
Index entries for sequences related to compositions

Cf. A028260, A286219, A286223, A286225, A286226.

nmax = 45; CoefficientList[Series[1/(1 - Sum[Boole[EvenQ[PrimeOmega[k]]] x^k, {k, 1, nmax}]), {x, 0, nmax}], x]

G.f.: 1/(1 - Sum_{k>=1} x^A028260(k)).

cp's OEIS Frontend

A286224 Number of compositions (ordered partitions) of n into parts with an odd number of distinct prime divisors.

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