A167197 a(6) = 7, for n >= 7, a(n) = a(n - 1) + gcd(n, a(n - 1)).

7, 14, 16, 17, 18, 19, 20, 21, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 52, 53, 54, 55, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 116, 117, 120, 121, 122, 123, 124, 125, 126, 127, 128

Offset: 6

Author

Vladimir Shevelev, Oct 30 2009, Nov 06 2009

Keywords

• nonn

Comments

For every n >= 7, a(n) - a(n - 1) is 1 or prime. This Rowland-like "generator of primes" is different from A106108 (see comment to A167168) and from A167170. Note that, lim sup a(n) / n = 2, while lim sup A106108(n) / n = lim sup A167170(n) / n = 3.

Going up to a million, differences of two consecutive terms of this sequence gives primes about 0.009% of the time. The rest are 1's. [Alonso del Arte, Nov 30 2009]

Links

• G. C. Greubel, Table of n, a(n) for n = 6..1000
• E. S. Rowland, A natural prime-generating recurrence, Journal of Integer Sequences, 11 (2008), Article 08.2.8.
• Vladimir Shevelev, An infinite set of generators of primes based on the Rowland idea and conjectures concerning twin primes, arXiv:0910.4676 [math.NT], 2009.

Crossrefs

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

Programs

Maple

A[6]:= 7:
for n from 7 to 100 do A[n]:= A[n-1] + igcd(n,A[n-1]) od:
seq(A[i],i=6..100); # Robert Israel, Jun 05 2016

Mathematica

a[6] = 7; a[n_ /; n > 6] := a[n] = a[n - 1] + GCD[n, a[n - 1]]; Table[a[n], {n, 6, 58}]

Python

from math import gcd
def aupton(nn):
    alst = [7]
    for n in range(7, nn+1): alst.append(alst[-1] + gcd(n, alst[-1]))
    return alst
print(aupton(68)) # Michael S. Branicky, Jul 14 2021

Extensions

Verified and edited by Alonso del Arte, Nov 30 2009