A001031 -id:A001031 - cp's OEIS frontend

A281423 Expansion of (Sum_{k>=1} x^prime(prime(k)))^2 [even terms only].

0, 0, 0, 1, 2, 1, 0, 2, 2, 0, 2, 3, 0, 0, 2, 0, 0, 3, 2, 0, 0, 2, 2, 2, 2, 0, 2, 0, 0, 2, 0, 3, 2, 0, 0, 4, 4, 0, 2, 2, 0, 1, 2, 2, 2, 2, 0, 2, 0, 2, 4, 0, 0, 0, 2, 0, 2, 4, 0, 1, 2, 0, 2, 4, 0, 2, 2, 1, 0, 2, 2, 2, 2, 0, 0, 4, 0, 0, 0, 2, 2, 2, 0, 1, 6, 0, 0, 2, 2, 0, 0, 2, 2, 2, 2, 2, 2, 4, 4, 2, 0, 2, 0, 0, 2, 4, 0, 2, 4, 1, 2, 4

Offset: 0

Ilya Gutkovskiy, Jan 21 2017

nonn

Number of ways to write 2n as an ordered sum of two primes with prime subscripts (A006450).

			a(4) = 2 because we have [3, 5] and [5, 3], where 3 = prime(2) = prime(prime(1)) and 5 = prime(3) = prime(prime(2)).

Ilya Gutkovskiy, Extended graphical example
Index entries for sequences related to compositions

Cf. A001031, A002375, A006450, A045917, A073610, A281422.

Take[CoefficientList[Series[Sum[x^Prime[Prime[k]], {k, 1, 250}]^2, {x, 0, 250}], x], {1, -1, 2}]

G.f.: (Sum_{k>=1} x^prime(prime(k)))^2 [even terms only].

A188766 Numbers n such that the number of decompositions of 2n into sum of two primes (counting 1 as a prime) is 1 or a composite.

Original entry on oeis.org

1, 12, 15, 17, 18, 22, 23, 24, 25, 27, 29, 31, 33, 37, 42, 44, 45, 46, 49, 50, 51, 52, 53, 54, 58, 59, 60, 61, 63, 64, 66, 67, 69, 70, 71, 73, 77, 79, 80, 81, 82, 83, 84, 85, 86, 87, 90, 92, 95, 96, 97, 98, 99, 100, 101, 102, 107, 110, 112, 115, 117, 118, 119

Offset: 1

Jonathan Vos Post, Apr 09 2011

Arises in Goldbach conjecture.

			1 is a term because there is a unique decomposition of 2*1 = 2 into a sum of two primes (counting 1 as a prime), namely 2 = 1 + 1.
12 is a term because there are 4 decompositions of 2*12 = 24 into a sum of two primes (counting 1 as a prime), namely 1 + 23, 5 + 19, 7 + 17, and 11 + 13, and 4 is a composite number.

Cf. A000040, A001031, A002808, A018252.

def is_A188766(n):
    pp = set(prime_range(2*n+1)+[1])
    return not is_prime(len([x for x in Partitions(2*n,length=2) if set(x) <= pp]))
# D. S. McNeil, Apr 10 2011

{integers n > 0 such that A001031(n) is in A018252} = {integers n > 0 such that A001031(n) is not in A000040}.

cp's OEIS Frontend

A281423 Expansion of (Sum_{k>=1} x^prime(prime(k)))^2 [even terms only].

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula