%I #24 Feb 28 2018 09:54:05

%S 0,1,3,11,58,57,262,261,564,666,665,4775,7843,7842,9807,9804,15705,

%T 15704,15701,15699,15698,77964,77962,180330,180326,185136,185135,

%U 185133,185132,185130,678603,678601,1005372,1005371,1005366,2326178,8525865,8525862,8525860

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

%C 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?

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

%C 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),...

%C From _Chai Wah Wu_, Feb 27 2018: (Start)

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

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

%C 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.

%C 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.

%C (End)

%H Chai Wah Wu, <a href="/A229512/b229512.txt">Table of n, a(n) for n = 1..240</a>

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

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

%Y Cf. A000040, A002386, A151800.

%K nonn

%O 1,3

%A _Vladimir Shevelev_, Sep 25 2013

%E More terms from _Peter J. C. Moses_