A229512 a(n) is the minimal k such that nextprime(2k+1) - 2k = prime(n) where nextprime(n) is least prime > n.

0, 1, 3, 11, 58, 57, 262, 261, 564, 666, 665, 4775, 7843, 7842, 9807, 9804, 15705, 15704, 15701, 15699, 15698, 77964, 77962, 180330, 180326, 185136, 185135, 185133, 185132, 185130, 678603, 678601, 1005372, 1005371, 1005366, 2326178, 8525865, 8525862, 8525860

Offset

1,3

Comments

It is interesting to note the distribution of terms over groups of close magnitudes. For example, 58,57; 9807,9804;, 15704,15701,15699,15698; etc. Is there an explanation?

If a(n+1) = a(n) - 1 and 2*a(n) + 1 is not prime, then (prime(n), prime(n+1)) is twin pair.

We call a twin pair (prime(n)), prime(n+1)) regular, if a(n+1) = a(n) - 1, and irregular otherwise. The first irregular pairs are (3,5),(5,7),(149,151),...

From Chai Wah Wu, Feb 27 2018: (Start)

a(n) exists for all n since the prime gap can be arbitrarily large.

If k > 0 is a term, then p <= 2k+1 < nextprime(p) for some prime p in A002386.

In this case, for k = (p-1)/2, nextprime(2*m+1) - 2*m = g - 2*(m-k) for k <= m < k + (g-1)/2 where g = nextprime(p) - p + 1.

For large g, this set of numbers m will include a few terms of the sequence (as g - 2*(m-k) will cover some primes not yet found) which result in the clusters of terms observed.

(End)

Links

Chai Wah Wu, Table of n, a(n) for n = 1..240

Mathematica

Table[Block[{k = 0}, While[NextPrime[2 k + 1] - 2 k != p, k++]; k], {p, Prime@ Range@ 30}] (* Michael De Vlieger, Feb 28 2018 *)

Prog

(PARI) a(n) = my(k = 0, p = prime(n)); while (nextprime(2*k+2) - 2*k != p, k++); k \\ Michel Marcus, Sep 25 2013

Crossrefs

Cf. A000040, A002386, A151800.

Sequence in context: A001586 A126201 A261643 * A208990 A020012 A126100

Adjacent sequences: A229509 A229510 A229511 * A229513 A229514 A229515

Keyword

nonn

Author

Vladimir Shevelev, Sep 25 2013

Extensions

More terms from Peter J. C. Moses

Status

approved