A191304 -id:A191304 - cp's OEIS frontend

A291153 a(n) is the prime index of A191304(n+1).

3, 5, 9, 15, 26, 51, 91, 160, 290, 526, 959, 1767, 3279, 6113, 11426, 21456, 40448, 76548, 145205, 276032, 526142, 1004977, 1924032, 3689162, 7086486, 13633821, 26269617, 50680636, 97899691, 189336057, 366569494, 710444878, 1378224063, 2676107406, 5200648226, 10114912373, 19687771058, 38348128843, 74746149884, 145785668141, 284517554507, 555594884599, 1085551499862, 2122142209034, 4150687469435

Offset: 1

Ralf Steiner, Aug 19 2017

nonn

The left point (x,y) of intersection of quadratic fits of log(a(n)) and log(A191304(n+1)) is about (-1, 0).
a(n+1) < 2 a(n) for all n, and lim_{n->inf} a(n+1)/a(n) = 2.
With A167168(1)=3 and s_1 = {3,5,11,23,...}, p_(a(n)) = s_1(n+1) in a two-index notation for every prime p_i for i > 1 based on Shevelev's equivalence classes of Rowland-like prime sequence recurrences. These equivalence classes {s_n(k)} were defined by Shevelev, see Crossrefs.

			p_(a(3)) = A000040(a(3)) = A000040(9) = 23 = s_1(3+1) with
s_1 = {3,5,11,23,...}.

Cf. A191304, A167168 (equivalence classes), A000040 (prime numbers).

Rest@ PrimePi@ Union@ FoldList[Max, 1, Rest@ # - Most@ #] &@ FoldList[#1 + GCD[#2, #1] &, 7, Range[2, 10^7]] (* after Michael De Vlieger, Aug 19 2017, after Robert G. Wilson v at A132199 *)

a(n) = pi(A191304(n+1)).

(4/5)^2 (n - 1) < log(a(n)) < (4/5)^2 (n + 1), for at least n < 46.

A132199 Rowland's prime-generating sequence: first differences of A106108.

Original entry on oeis.org

1, 1, 1, 5, 3, 1, 1, 1, 1, 11, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 23, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 47, 3, 1, 5, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 101, 3, 1, 1

Offset: 1

N. J. A. Sloane, Jan 28 2008

nonn

Rowland shows that the terms are all 1's or primes.
The prime terms form A137613.
See A137613 for additional comments, links and references. - Jonathan Sondow, Aug 14 2008
From Robert G. Wilson v, Apr 30 2009: (Start)
First appearance of k-th prime, k >= 0: 1, 0, 5, 4, 104, 10, 116, 242878, 242819, 22, 243019, 3891770, 242867, ..., .
The number of different numbers in the first 10^k terms beginning with k=0: 1, 4, 7, 12, 15, 19, 30, >35, ..., .
Record high values are A191304. (End)

Eric S. Rowland, A simple prime-generating recurrence, Abstracts Amer. Math. Soc., 29 (No. 1, 2008), p. 50 (Abstract 1035-11-986).

Robert G. Wilson v, Table of n, a(n) for n = 1..10000 .
Jean-Paul Delahaye, Déconcertantes conjectures, Pour la Science (French edition of Scientific American), No. 367, May 2008.
Brian Hayes, Pumping the Primes, bit-player, 19 August 2015.
Mario Raso, Integer Sequences in Cryptography: A New Generalized Family and its Application, Ph. D. Thesis, Sapienza University of Rome (Italy 2025). See p. 18.
Eric S. Rowland, A natural prime-generating recurrence, arXiv:0710.3217 [math.NT], 2007-2008.
Eric S. Rowland, A simple prime-generating recurrence, 2008.
Eric S. Rowland, Prime-Generating Recurrences and a Tale of Logarithmic Scale, 20 January 2023.

Cf. A106108, A137613, A134734, A134743, A134744, A191304 (record highs) A247090. See A106108 for other cross-references.

a132199 n = a132199_list !! (n-1)
a132199_list = zipWith (-) (tail a106108_list) a106108_list
-- Reinhard Zumkeller, Nov 15 2013

A106108 := proc(n)
    option remember;
    if n = 1 then
        7;
    else
        procname(n-1)+igcd(n,procname(n-1)) ;
    end if;
end proc:
A132199 := proc(n)
    A106108(n+1)-A106108(n) ;
end proc: # R. J. Mathar, Jul 04 2013

a[1] = 7; a[n_] := a[n] = a[n - 1] + GCD[n, a[n - 1]]; t = Array[a, 104]; Rest@t - Most@t (* Robert G. Wilson v, Apr 30 2009 *)

ub=1000; a=7; n=2; while(nDaniel Constantin Mayer, Aug 31 2014

from itertools import count, islice
from math import gcd
def A132199_gen(): # generator of terms
    a = 7
    for n in count(2):
        yield (b:=gcd(a,n))
        a += b
A132199_list = list(islice(A132199_gen(),20)) # Chai Wah Wu, Mar 14 2023

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A291153 a(n) is the prime index of A191304(n+1).

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A132199 Rowland's prime-generating sequence: first differences of A106108.

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