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

1,2

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.

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.

Formula

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

Example

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.

Mathematica

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

Prog

(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)

Crossrefs

Cf. A000326, A002822, A001359, A024702.

Sequence in context: A136113 A298992 A050189 * A116995 A032281 A294654

Adjacent sequences: A308341 A308342 A308343 * A308345 A308346 A308347

Keyword

nonn

Author

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

Status

approved