A286222 -id:A286222 - cp's OEIS frontend

A286223 Number of partitions of n into distinct parts with an even number of prime divisors (counted with multiplicity).

1, 1, 0, 0, 1, 1, 1, 1, 0, 1, 3, 2, 0, 1, 3, 4, 4, 2, 1, 4, 6, 5, 4, 4, 6, 10, 10, 6, 6, 10, 13, 14, 11, 9, 14, 21, 21, 17, 17, 23, 31, 31, 25, 25, 33, 41, 43, 39, 38, 50, 61, 60, 56, 58, 68, 83, 87, 79, 82, 99, 115, 121, 118, 118, 139, 163, 164, 157, 165, 189, 216, 228, 221, 229, 265, 296

Offset: 0

Ilya Gutkovskiy, May 04 2017

nonn

			a(10) = 3 because we have [10], [9, 1] and [6, 4].

Amiram Eldar, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, Prime Factor
Index entries for related partition-counting sequences

Cf. A028260, A286219, A286221, A286222.

nmax = 75; CoefficientList[Series[Product[1 + Boole[EvenQ[PrimeOmega[k]]] x^k, {k, 1, nmax}], {x, 0, nmax}], x]

G.f.: Product_{k>=1} (1 + x^A028260(k)).

A286226 Number of compositions (ordered partitions) of n into parts with an odd number of prime divisors (counted with multiplicity).

Original entry on oeis.org

1, 0, 1, 1, 1, 3, 2, 6, 7, 10, 18, 22, 39, 54, 81, 127, 179, 282, 412, 620, 940, 1384, 2106, 3129, 4698, 7062, 10531, 15856, 23695, 35541, 53304, 79773, 119723, 179262, 268748, 402832, 603484, 904821, 1355652, 2031963, 3045374, 4563604, 6840316, 10250612, 15363248, 23024904, 34506390

Offset: 0

Ilya Gutkovskiy, May 04 2017

nonn

			a(5) = 3 because we have [5], [3, 2] and [2, 3].

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. A026424, A286218, A286222, A286224, A286227.

nmax = 46; CoefficientList[Series[1/(1 - Sum[Boole[OddQ[PrimeOmega[k]]] x^k, {k, 1, nmax}]), {x, 0, nmax}], x]

G.f.: 1/(1 - Sum_{k>=1} x^A026424(k)).

cp's OEIS Frontend

A286223 Number of partitions of n into distinct parts with an even number of prime divisors (counted with multiplicity).

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