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.
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
CROSSREFS
KEYWORD
nonn
AUTHOR
M. F. Hasler and A. Dinculescu, Jul 04 2019
STATUS
approved