A276034 a(n) is the number of decompositions of 2n into an unordered sum of two primes in A274987.

0, 0, 1, 1, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 0, 3, 2, 1, 2, 2, 2, 1, 2, 1, 0, 2, 1, 1, 2, 2, 3, 3, 2, 2, 2, 2, 3, 2, 1, 2, 4, 3, 1, 5, 3, 2, 5, 1, 2, 2, 2, 5, 2, 3, 4, 5, 3, 2, 5, 2, 1, 4, 0, 1, 5, 3, 1, 3, 5, 4, 4, 3, 2, 4, 3, 3, 4, 2, 3, 7, 2, 2, 3, 2, 2, 2

Offset

1,5

Comments

The two primes are allowed to be the same.

It is conjectured that the primes in A274987 (a subset of all primes) are sufficient to decomposite even numbers into two primes in A274987 when n > 958.

This sequence provides a very tight alternative of the Goldbach conjecture for all positive integers, in which indices of zero terms form a complete sequence {1, 2, 16, 26, 64, 97, 107, 122, 146, 167, 194, 391, 451, 496, 707, 856, 958}.

There is no more zero terms of a(n) tested up to n = 100000.

Links

Lei Zhou, Table of n, a(n) for n = 1..10000

Lei Zhou, List plot of the first 10000 terms of a(n).

Example

A274987 = {3, 5, 7, 11, 13, 17, 23, 31, 37, 53, 59, 61, 73, 79, 83, 89, 101, 103, 109, ...}.
For n=3, 2n=6 = 3+3, one case of decomposition, so a(3)=1;
for n=4, 2n=8 = 3+5, one case of decomposition, so a(4)=1;
...
for n=17, 2n=34 = 3+31 = 11+23 = 17+17, three cases of decompositions, so a(17)=3.

Mathematica

p = 3; sp = {p}; a = Table[m = 2*n; l = Length[sp]; While[sp[[l]] < m, While[p = NextPrime[p]; cp = 2*3^(Floor[Log[3, 2*p - 1]]) - p; ! PrimeQ[cp]]; AppendTo[sp, p]; l++]; ct = 0; Do[If[(2*sp[[i]] <= m) && (MemberQ[sp, m - sp[[i]]]), ct++], {i, 1, l}]; ct, {n, 1, 87}]

Crossrefs

Cf. A002375, A045917, A001031, A274987, A171611, A240708, A240712, A230443.

Sequence in context: A297776 A043535 A043560 * A111626 A297777 A043536

Adjacent sequences: A276031 A276032 A276033 * A276035 A276036 A276037

Keyword

nonn,easy,base,look

Author

Lei Zhou, Nov 15 2016

Status

approved