A394657 Number of decompositions of 2n into an unordered sum of two terms of A393798.

0, 1, 1, 1, 2, 1, 2, 2, 2, 2, 3, 3, 3, 2, 3, 2, 4, 4, 2, 3, 4, 3, 4, 5, 4, 3, 5, 3, 4, 6, 3, 5, 6, 2, 5, 6, 5, 5, 7, 4, 5, 8, 5, 4, 9, 4, 5, 7, 3, 6, 8, 5, 6, 8, 6, 7, 10, 6, 6, 12, 4, 5, 10, 3, 6, 8, 5, 5, 7, 6, 8, 10, 5, 5, 11, 3, 7, 9, 4, 7, 9, 4, 5, 11, 8

Offset

1,5

Comments

The second different term of this sequence to A002375 is a(65).

Conjecture: for n >= 3, this sequence is always positive.

This is a stronger version of the Goldbach Conjecture.

Links

Table of n, a(n) for n=1..85.

Example

For n = 1, 2n = 2, which cannot be presented as the sum of two primes.
For n = 2, 2n = 4 = 2 + 2, thus a(2) = 1.
For 3 <= n <= 64, refer to examples in A002375.
For n = 65, 2n = 130. A393798 up to 130 gives {2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113}. We have 130 = 17+113 = 23+107 = 29+101 = 41+89 = 47+83 = 59+71. Six instances found. So a(65)=6.
Where as for A002375, there is one more instance as 3+127, however 127 is not a term in A393798.

Mathematica

a393798 = {2}; Table[s = 2*n; While[a393798[[-1]] < s, p = a393798[[-1]]; q = p; While[q = NextPrime[q]; diff = q - p; f = FactorInteger[diff]; l = Length[f]; f[[l, 1]] > 3]; p = q; AppendTo[a393798, p]]; pos = 0; ct = 0; While[pos++; pos <= Length[a393798], p = a393798[[pos]]; If[p <= n, If[MemberQ[a393798, s - p], ct++]]]; ct, {n, 1, 85}]

Crossrefs

Cf. A000040, A002375, A045917, A240708, A254041, A319178, A393798.

Sequence in context: A332656 A230443 A254610 * A002375 A045917 A240708

Adjacent sequences: A394654 A394655 A394656 * A394658 A394659 A394660

Keyword

nonn,easy

Author

Lei Zhou, Mar 27 2026

Status

approved