A213536 Cousin prime recurrence sequence: a(1) = 14, and for n > 1, a(n) = a(n-1) + gcd(n+5, a(n-1)), if n is even, otherwise a(n) = a(n-1) + gcd(n+1, a(n-1)).

14, 21, 22, 23, 24, 25, 26, 39, 40, 45, 48, 49, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 111, 112, 113, 114, 115, 116, 117, 118, 119, 126, 127, 128, 129, 130, 131, 132, 133, 134, 135, 136, 137, 138, 139, 140, 141, 142, 143

Offset

1,1

Comments

Conjecture: Record differences a(n) - a(n-1) (A213537) are a strict subset of the smaller of cousin primes (A023200). (Cousin primes differ by 4.)

Conjecture: Record differences are an infinite sequence. It is widely believed there are infinitely many cousin primes. (Similarly, by Dickson's conjecture and the second Hardy-Littlewood conjecture, there are infinitely many pairs of (not necessarily consecutive) primes (p, p+2k) for each natural number k.)

Conjecture: The following pattern makes sequences for every (necessarily even) difference (slight change for 2). For difference d, p is first prime > d that is the smaller of a prime pair (p, p+d). a(1) = 2p and a(n) = gcd(n+p-2, a(n-1)) for even n, otherwise gcd(n+p-2-d, a(n-1)).

Links

T. D. Noe, Table of n, a(n) for n = 1..1273

G. H. Hardy and J. E. Littlewood, Some problems of 'Partitio numerorum'; III: on the expression of a number as a sum of primes, Acta Mathematica, Vol. 44, pp. 1-70, 1923.

E. S. Rowland, A natural prime-generating recurrence, Journal of Integer Sequences, Vol. 11 (2008), Article 08.2.8.

Pascal Sebah and Xavier Gourdon, Introduction to twin primes and Brun's constant.

Vladimir Shevelev, A new generator of primes based on the Rowland idea, arXiv:0910.4676 [math.NT], 2009.

Vladimir Shevelev, Three theorems on twin primes, arXiv:0911.5478 [math.NT], 2009-2010.

Wikipedia, Dickson's conjecture.

Maple

A213536 := proc(n)
    option remember;
    if n = 1 then
        14;
    elif type(n, 'even') then
        procname(n-1)+gcd(n+5, procname(n-1)) ;
    else
        procname(n-1)+gcd(n+1, procname(n-1)) ;
    end if;
end proc: # R. J. Mathar, Jul 20 2012

Mathematica

nxt[{n_, a_}]:={n+1, If[OddQ[n], a+GCD[a, n+6], a+GCD[a, n+2]]}; Transpose[ NestList[nxt, {1, 14}, 60]][[2]] (* Harvey P. Dale, Jun 22 2013 *)

Crossrefs

Cf. A213537 (record differences).

Sequence in context: A303475 A028395 A167170 * A330137 A230095 A133185

Adjacent sequences: A213533 A213534 A213535 * A213537 A213538 A213539

Keyword

nonn

Author

Joseph Benstock, Jun 27 2012

Status

approved