A361919 The number of primes > A000040(n) and <= (A000040(n)^c + 1)^(1/c), where c = 0.567148130202... is defined in A038458.

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, 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, 4, 3, 3, 3, 4, 3, 4, 3, 3, 3, 3, 4, 5, 4, 4, 4, 5, 5

Offset: 1

Author

Hal M. Switkay, Mar 29 2023

Keywords

• nonn

Comments

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.

Examples

			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.
		

Links

• Hal M. Switkay, Table of n, a(n) for n = 1..665
• F. Smarandache, Conjectures which generalize Andrica's conjecture, arXiv:0707.2584 [math.GM], 2007; Octogon 7:1 (1999), pp. 173-176.

Crossrefs

Cf. A000040, A038458.