A308344 a(n) = (A001359(n+1)^2 - 1)/24, where A001359 = lesser of twin primes; or: pentagonal numbers (A000326) whose indices are twin ranks (A002822).

1, 5, 12, 35, 70, 145, 210, 425, 477, 782, 925, 1335, 1520, 1617, 2147, 2380, 3015, 3290, 4030, 5017, 7315, 7740, 8855, 11310, 13490, 14950, 15862, 17120, 18095, 27270, 28085, 28497, 30602, 32340, 43265, 44290, 45850, 46905, 49595, 55200, 62935, 67947, 69230, 70525

Offset: 1

Author

M. F. Hasler and A. Dinculescu, Jul 04 2019

Keywords

• nonn

Comments

Subsequence of A024702 which considers all primes rather than only twins.

This sequence seems to play an important role in studying the twin prime conjecture; see also A057767, A273257, and related.

Dinculescu calls the numbers M(j) = (prime(j)^2 - 1)/6 "basic numbers", and [M(j), M(j+1)] a "twin interval" when j is the index of a twin prime. He notes that the length of such an interval equals four times the corresponding twin rank k(j) = (prime(j) + prime(j+1))/6, see near eq.(3.3) in the 2018 paper.

Examples

			Sequence A001359 = {3, 5, 11, 17, 29, ...} lists the lesser members of pairs of twin primes, (3, 5), (5, 7), (11, 13), (17, 19), ...
We ignore the first and start with the second pair, (5, 7). We have (5^2 - 1)/24 = 1 = a(1).
Next comes the pair (11, 13), whence (11^2 - 1)/24 = 120/24 = 5 = a(2), etc.
		

Links

• M. F. Hasler, Table of n, a(n) for n = 1..10000
• A. Dinculescu, On the Numbers that Determine the Distribution of Twin Primes, Surveys in Mathematics and its Applications, 13 (2018), 171-181.

Crossrefs

Cf. A000326, A002822, A001359, A024702.

Programs

Mathematica

(#^2-1)/24&/@Rest[Select[Partition[Prime[Range[300]],2,1],#[[2]]-#[[1]] == 2&][[All,1]]] (* Harvey P. Dale, Sep 05 2020 *)

PARI

a(n)=A000326(A002822(n))

PARI

a(n)=(A001359(n+1)^2-1)/24 \\ or implemented as follows:
p=0;forprime(q=5,oo,p+2==q&&print1(p^2\24",");p=q)

Formula

a(n) = (A001359(n+1)^2 - 1)/24 = A000326(A002822(n)).