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A308344
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a(n) = (A001359(n+1)^2 - 1)/24, where A001359 = lesser of twin primes; or: pentagonal numbers (A000326) whose indices are twin ranks (A002822).
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2
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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
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OFFSET
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1,2
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COMMENTS
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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.
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LINKS
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FORMULA
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EXAMPLE
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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.
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MATHEMATICA
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(#^2-1)/24&/@Rest[Select[Partition[Prime[Range[300]], 2, 1], #[[2]]-#[[1]] == 2&][[All, 1]]] (* Harvey P. Dale, Sep 05 2020 *)
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PROG
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(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)
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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