A337772 Number of Goldbach partitions (p,q) of 2n such that any one of p+-1 or q+-1 is a perfect square.

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

Offset

1,5

Links

Table of n, a(n) for n=1..93.

Eric Weisstein's World of Mathematics, Goldbach Partition

Wikipedia, Goldbach's conjecture

Index entries for sequences related to Goldbach conjecture

Index entries for sequences related to partitions

Formula

a(n) = Sum_{i=1..n} sign(s(i-1) + s(i+1) + s(2*n-i-1) + s(2*n-i+1)) * c(i) * c(2*n-i), where s is the square characteristic (A010052) and c is the prime characteristic (A010051).

Example

a(5) = 2; 2*5 = 10 has two Goldbach partitions, (7,3) and (5,5). Since 3+1 = 4 is a square and 5-1 = 4 is a square, a(5) = 2.
a(17) = 3; 2*17 = 34 has the four Goldbach partitions, (31,3), (29,5), (23,11) and (17,17). Since 3+1 = 4 (square), 5-1 = 4 (square), and 17-1 = 16 (square), a(17) = 3.

Mathematica

Table[Sum[Sign[(Floor[Sqrt[i - 1]] - Floor[Sqrt[i - 2]]) + (Floor[Sqrt[2 n - i - 1]] - Floor[Sqrt[2 n - i - 2]]) + (Floor[Sqrt[i + 1]] - Floor[Sqrt[i]]) + (Floor[Sqrt[2 n - i + 1]] - Floor[Sqrt[2 n - i]])] * (PrimePi[i] - PrimePi[i - 1]) (PrimePi[2 n - i] - PrimePi[2 n - i - 1]), {i, n}], {n, 100}]

Crossrefs

Cf. A010051, A010052, A045917.

Sequence in context: A175128 A359307 A205154 * A371243 A025898 A025849

Adjacent sequences: A337769 A337770 A337771 * A337773 A337774 A337775

Keyword

nonn

Author

Wesley Ivan Hurt, Sep 19 2020

Status

approved