A346777 a(n) is the number of consecutive even prime gaps (g1, g2) satisfying g1 == 2 (mod 6) and g2 == 4 (mod 6) out of the first 2^n consecutive even prime gaps.

0, 1, 2, 3, 4, 9, 16, 27, 56, 111, 187, 373, 708, 1403, 2780, 5467, 10781, 21248, 41701, 82581, 163473, 323995, 643327, 1278401, 2540048, 5050955, 10052647, 20010073

Offset

0,3

Comments

The prime gaps are given in A001223. Here we consider the gaps satisfying the conditions A001223(k) == 2 and A001223(k+1) == 4 (mod 6) for 1 < k <= 2^n + 1.

Links

Table of n, a(n) for n=0..27.

Formula

a(n) = A341531(n) - A346776(n) - 1.

Example

The sequence A001223(n) mod 6 is given by:
1, 2, 2, 4, 2, 4, 2, 4, 0, 2, 0, 4, 2, 4, 0, 0, 2, 0, 4, 2, 0, 4, 0, 2, ..., denoted here as b(0), b(1), b(2), ..., i.e. b(n) = A001223(n+1) (mod 6) for n >= 0.
The term b(0) is excluded by definition. The conditions b(k) = 2 and b(k+1) == 4 are obtained for k = 2, 4, 6, 12 ...
So a(0) = 0 (k = 2^0 does not occur), a(1) = 1 (one value of k satisfying k <= 2^1), a(2) = 2 (two value of k satisfying k <= 2^2) and a(3) = 3 (three value of k satisfying k <= 2^3).

Crossrefs

Cf. A001223, A340948, A341531, A341532, A345332, A345333, A345334, A346776.

Sequence in context: A354268 A281882 A325436 * A338585 A145027 A088275

Adjacent sequences: A346774 A346775 A346776 * A346778 A346779 A346780

Keyword

nonn,more

Author

A.H.M. Smeets, Aug 03 2021

Status

approved