A166944 a(1)=2; a(n) = a(n-1) + gcd(n, a(n-1)) if n is even, a(n) = a(n-1) + gcd(n-2, a(n-1)) if n is odd.

2, 4, 5, 6, 9, 12, 13, 14, 21, 22, 23, 24, 25, 26, 39, 40, 45, 54, 55, 60, 61, 62, 63, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 129, 130, 135, 138, 139, 140, 147, 148, 149, 150, 151, 152, 153, 154, 155, 160, 161, 162, 163

Offset: 1

Author

Vladimir Shevelev, Oct 24 2009

Keywords

• nonn

Comments

Conjecture: Every record of differences a(n)-a(n-1) more than 5 is the greater of twin primes (A006512).

Links

• Harvey P. Dale, Table of n, a(n) for n = 1..1000
• E. S. Rowland, A natural prime-generating recurrence, Journal of Integer Sequences, Vol.11(2008), Article 08.2.8. arXiv:0710.3217 [math.NT]
• V. Shevelev, An infinite set of generators of primes based on the Rowland idea and conjectures concerning twin primes, arXiv:0910.4676 [math.NT], 2009. - _Vladimir Shevelev_, Oct 27 2009
• V. Shevelev, Three theorems on twin primes, arXiv:0911.5478 [math.NT], 2009-2010. - _Vladimir Shevelev_, Dec 03 2009

Crossrefs

Cf. A084662, A084663, A106108, A132199, A134162, A135506, A135508, A118679, A120293.

Programs

Maple

A166944 := proc(n) option remember; if n = 1 then 2; else p := procname(n-1) ; if type(n,'even') then p+igcd(n,p) ; else p+igcd(n-2,p) ; end if; end if; end proc: # R. J. Mathar, Sep 03 2011

Mathematica

nxt[{n_,a_}]:={n+1,If[OddQ[n],a+GCD[n+1,a],a+GCD[n-1,a]]}; Transpose[ NestList[ nxt,{1,2},70]][[2]] (* Harvey P. Dale, Feb 10 2015 *)

PARI

print1(a=2); for(n=2, 100, d=gcd(a, if(n%2, n-2, n)); print1(", "a+=d)) \\ Charles R Greathouse IV, Oct 13 2017

Extensions

Terms beginning with a(18) corrected by Vladimir Shevelev, Nov 10 2009