A361919 - OEIS

%I #13 Apr 01 2023 23:32:03

%S 1,1,1,1,1,1,2,1,1,1,1,2,2,1,1,2,2,2,2,2,2,2,1,2,4,4,3,2,1,1,3,2,3,2,

%T 3,2,3,3,3,2,3,3,3,2,2,1,3,5,4,3,3,3,2,3,3,4,4,3,3,2,1,3,3,3,2,2,4,4,

%U 4,3,3,3,4,3,4,3,3,3,3,4,5,4,4,4,5,5

%N The number of primes > A000040(n) and <= (A000040(n)^c + 1)^(1/c), where c = 0.567148130202... is defined in A038458.

%C Let c = 0.567148130202... (see A038458), the solution to 127^x - 113^x = 1. c is conjectured by Smarandache to be the smallest real number x such that A000040(n+1)^x - A000040(n)^x = 1 has a solution. This conjecture is equivalent to saying that the terms of the present sequence are always positive, but that if c were replaced by a larger real number, there would be zeros in the sequence. However, note that a(30) is not the last occurrence of 1: a(46) = a(61) = 1 as well.

%H Hal M. Switkay, <a href="/A361919/b361919.txt">Table of n, a(n) for n = 1..665</a>

%H F. Smarandache, <a href="http://arxiv.org/abs/0707.2584">Conjectures which generalize Andrica's conjecture</a>, arXiv:0707.2584 [math.GM], 2007; Octogon 7:1 (1999), pp. 173-176.

%e a(30) is the number of primes > A000040(30), which is 113, and <= (113^c + 1)^(1/c) = 127. This relatively large interval contains only the prime 127.

%Y Cf. A000040, A038458.

%K nonn

%O 1,7

%A _Hal M. Switkay_, Mar 29 2023