This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A308344 #24 Sep 05 2020 12:12:02
%S A308344 1,5,12,35,70,145,210,425,477,782,925,1335,1520,1617,2147,2380,3015,
%T A308344 3290,4030,5017,7315,7740,8855,11310,13490,14950,15862,17120,18095,
%U A308344 27270,28085,28497,30602,32340,43265,44290,45850,46905,49595,55200,62935,67947,69230,70525
%N 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).
%C A308344 Subsequence of A024702 which considers all primes rather than only twins.
%C A308344 This sequence seems to play an important role in studying the twin prime conjecture; see also A057767, A273257, and related.
%C A308344 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.
%H A308344 M. F. Hasler, <a href="/A308344/b308344.txt">Table of n, a(n) for n = 1..10000</a>
%H A308344 A. Dinculescu, <a href="http://www.utgjiu.ro/math/sma/v13/p13_11.pdf">On the Numbers that Determine the Distribution of Twin Primes</a>, Surveys in Mathematics and its Applications, 13 (2018), 171-181.
%F A308344 a(n) = (A001359(n+1)^2 - 1)/24 = A000326(A002822(n)).
%e A308344 Sequence A001359 = {3, 5, 11, 17, 29, ...} lists the lesser members of pairs of twin primes, (3, 5), (5, 7), (11, 13), (17, 19), ...
%e A308344 We ignore the first and start with the second pair, (5, 7). We have (5^2 - 1)/24 = 1 = a(1).
%e A308344 Next comes the pair (11, 13), whence (11^2 - 1)/24 = 120/24 = 5 = a(2), etc.
%t A308344 (#^2-1)/24&/@Rest[Select[Partition[Prime[Range[300]],2,1],#[[2]]-#[[1]] == 2&][[All,1]]] (* _Harvey P. Dale_, Sep 05 2020 *)
%o A308344 (PARI) a(n)=A000326(A002822(n))
%o A308344 (PARI) a(n)=(A001359(n+1)^2-1)/24 \\ or implemented as follows:
%o A308344 p=0;forprime(q=5,oo,p+2==q&&print1(p^2\24",");p=q)
%Y A308344 Cf. A000326, A002822, A001359, A024702.
%K A308344 nonn
%O A308344 1,2
%A A308344 _M. F. Hasler_ and A. Dinculescu, Jul 04 2019