A167495 - OEIS

%I #15 Dec 09 2018 14:41:56

%S 2,3,5,13,31,61,139,283,571,1153,2311,4651,9343,19141,38569,77419,

%T 154873,310231,621631,1243483,2486971,4974721

%N Records in A167494.

%C Conjecture: each term > 3 of the sequence is the greater member of a twin prime pair (A006512).

%C Indices of the records are 1, 2, 4, 6, 9, 10, 15, 18, 21, 25, 28, 30, 38, 72, 90, ... [_R. J. Mathar_, Nov 05 2009]

%C One can formulate a similar conjecture without verification of the primality of the terms (see Conjecture 4 in my paper). [_Vladimir Shevelev_, Nov 13 2009]

%H E. S. Rowland, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL11/Rowland/rowland21.html">A natural prime-generating recurrence</a>, Journal of Integer Sequences, Vol.11 (2008), Article 08.2.8.

%H E. S. Rowland, <a href="https://arxiv.org/abs/0710.3217">A natural prime-generating recurrence</a>, arXiv:0710.3217 [math.NT], 2007-2008.

%H V. Shevelev, <a href="https://arxiv.org/abs/0910.4676">A new generator of primes based on the Rowland idea</a>, arXiv:0910.4676 [math.NT], 2009.

%H V. Shevelev, <a href="https://arxiv.org/abs/0911.5478">Three theorems on twin primes</a>, arXiv:0911.5478 [math.NT], 2009-2010. [_Vladimir Shevelev_, Dec 03 2009]

%t nxt[{n_, a_}] := {n + 1, If[EvenQ[n], a + GCD[n+1, a], a + GCD[n-1, a]]};

%t A167494 = DeleteCases[Differences[Transpose[NestList[nxt, {1, 2}, 10^7]][[2]]], 1];

%t Tally[A167494][[All, 1]] //. {a1___, a2_, a3___, a4_, a5___} /; a4 <= a2 :> {a1, a2, a3, a5} (* _Jean-François Alcover_, Oct 29 2018, using _Harvey P. Dale_'s code for A167494 *)

%Y Cf. A167494, A167493, A167197, A167195, A167170, A167168, A106108, A132199, A167054, A167053, A166944, A166945, A116533, A163961, A163963, A084662, A084663, A134162, A135506, A135508, A118679, A120293.

%K nonn,more

%O 1,1

%A _Vladimir Shevelev_, Nov 05 2009

%E Simplified the definition to include all records; one term added by _R. J. Mathar_, Nov 05 2009

%E a(16) to a(21) from _R. J. Mathar_, Nov 19 2009

%E a(22) from _Jean-François Alcover_, Oct 29 2018