OFFSET
1,1
COMMENTS
Conjectures. 1) For n >= 2, every difference a(n) - a(n-1) is 1 or prime; 2) Every record of differences a(n) - a(n-1) greater than 3 belongs to the sequence of the greater of twin primes (A006512).
Conjecture #1 above fails at n = 620757, with a(n) = 1241487 and a(n-1) = 1241460, difference = 27. Additionally, the terms of related A167495(m) quickly tend to index n/2. So for example, A167495(14) = 19141 is seen at n = 38284. - Bill McEachen, Jan 20 2023
It seems that, for n > 4, (3*n-3)/2 <= a(n) <= 2n - 3. Can anyone find a proof or disproof? - Charles R Greathouse IV, Jan 22 2023
LINKS
Bill McEachen, Table of n, a(n) for n = 1..100000
E. S. Rowland, A natural prime-generating recurrence, Journal of Integer Sequences, Vol.11 (2008), Article 08.2.8.
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.
FORMULA
For n > 3, n < a(n) < n*(n-1)/2. - Charles R Greathouse IV, Jan 22 2023
MATHEMATICA
nxt[{n_, a_}]:={n+1, If[EvenQ[n], a+GCD[n+1, a], a+GCD[n-1, a]]}; Transpose[ NestList[nxt, {1, 2}, 70]][[2]] (* Harvey P. Dale, Dec 05 2015 *)
PROG
(PARI) lista(nn)=my(va = vector(nn)); va[1] = 2; for (n=2, nn, va[n] = if (n%2, va[n-1] + gcd(n, va[n-1]), va[n-1] + gcd(n-2, va[n-1])); ); va; \\ Michel Marcus, Dec 13 2018
(Python)
from math import gcd
from itertools import count, islice
def agen(): # generator of terms
an = 2
for n in count(2):
yield an
an = an + gcd(n, an) if n&1 else an + gcd(n-2, an)
print(list(islice(agen(), 66))) # Michael S. Branicky, Jan 22 2023
CROSSREFS
KEYWORD
nonn
AUTHOR
Vladimir Shevelev, Nov 05 2009
EXTENSIONS
More terms from Harvey P. Dale, Dec 05 2015
STATUS
approved