A338826 -id:A338826 - cp's OEIS frontend

A339380 Number of partitions of n into an even number of primes (counting 1 as a prime).

1, 0, 1, 1, 3, 2, 5, 4, 9, 7, 14, 11, 22, 18, 33, 27, 48, 40, 69, 58, 97, 82, 134, 114, 183, 157, 246, 212, 327, 284, 431, 376, 562, 493, 728, 640, 934, 825, 1191, 1056, 1508, 1341, 1899, 1694, 2377, 2126, 2960, 2654, 3668, 3297, 4523, 4075, 5554, 5015, 6792, 6145

Offset: 0

Ilya Gutkovskiy, Dec 02 2020

nonn

			a(6) = 5 because we have [5, 1], [3, 3], [3, 1, 1, 1], [2, 2, 1, 1] and [1, 1, 1, 1, 1, 1].

Index entries for sequences related to partitions

Cf. A000607, A008578, A027187, A034891, A184171, A184172, A184198, A184199, A338826, A339381, A339382, A339383.

b:= proc(n, i, t) option remember; (p->
      `if`(n=0, t, `if`(i<0, 0, b(n, i-1, t)+
      `if`(p>n, 0, b(n-p, i, 1-t)))))(`if`(i<1, 1, ithprime(i)))
    end:
a:= n-> b(n, numtheory[pi](n), 1):
seq(a(n), n=0..60);  # Alois P. Heinz, Dec 02 2020

nmax = 55; CoefficientList[Series[(1/2) ((1/(1 - x)) Product[1/(1 - x^Prime[k]), {k, 1, nmax}] + (1/(1 + x)) Product[1/(1 + x^Prime[k]), {k, 1, nmax}]), {x, 0, nmax}], x]
Table[Count[(Boole[PrimeQ/@(IntegerPartitions[n]/.(1->2))]),?(EvenQ[Length[#]] && FreeQ[ #,0]&)],{n,0,60}] (* _Harvey P. Dale, Aug 20 2024 *)

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

a(n) = (A034891(n) + A338826(n)) / 2.

A339381 Number of partitions of n into an odd number of primes (counting 1 as a prime).

Original entry on oeis.org

0, 1, 1, 2, 1, 4, 3, 7, 5, 11, 9, 18, 14, 27, 22, 40, 33, 58, 48, 82, 69, 114, 97, 157, 134, 212, 183, 284, 246, 376, 327, 493, 431, 640, 562, 825, 728, 1056, 934, 1341, 1191, 1694, 1508, 2126, 1899, 2654, 2377, 3297, 2960, 4075, 3668, 5015, 4523, 6145, 5554, 7499

Offset: 0

Ilya Gutkovskiy, Dec 02 2020

nonn

			a(6) = 3 because we have [3, 2, 1], [2, 2, 2] and [2, 1, 1, 1, 1].

Index entries for sequences related to partitions

Cf. A000607, A008578, A027193, A034891, A184171, A184172, A184198, A184199, A338826, A339380, A339382, A339383.

b:= proc(n, i, t) option remember; (p->
      `if`(n=0, t, `if`(i<0, 0, b(n, i-1, t)+
      `if`(p>n, 0, b(n-p, i, 1-t)))))(`if`(i<1, 1, ithprime(i)))
    end:
a:= n-> b(n, numtheory[pi](n), 0):
seq(a(n), n=0..60);  # Alois P. Heinz, Dec 02 2020

nmax = 55; CoefficientList[Series[(1/2) ((1/(1 - x)) Product[1/(1 - x^Prime[k]), {k, 1, nmax}] - (1/(1 + x)) Product[1/(1 + x^Prime[k]), {k, 1, nmax}]), {x, 0, nmax}], x]

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

a(n) = (A034891(n) - A338826(n)) / 2.

cp's OEIS Frontend

A339380 Number of partitions of n into an even number of primes (counting 1 as a prime).

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