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

Author

Ralf Steiner, Aug 19 2017

Keywords

• nonn

Comments

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.

Examples

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

Crossrefs

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

Programs

Mathematica

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 *)

Formula

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

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