A246776 a(n) = floor(prime(n)^(1+1/n)) - prime(n+1).

1, 0, 1, 0, 4, 2, 6, 4, 3, 9, 5, 8, 11, 9, 7, 8, 13, 9, 12, 14, 10, 13, 11, 10, 15, 17, 15, 17, 15, 5, 17, 15, 20, 11, 20, 16, 16, 19, 17, 17, 22, 13, 22, 20, 22, 12, 13, 22, 24, 22, 20, 24, 16, 21, 21, 21, 25, 21, 23, 25, 17, 14, 25, 27, 24, 14, 23, 20, 28, 26

Offset: 1

Author

Farideh Firoozbakht, Sep 26 2014

Keywords

• nonn

Comments

The Firoozbakht Conjecture, "prime(n)^(1/n) is a strictly decreasing function of n" is true if and only if a(n) is nonnegative for all n, n>1.

A246777 is a hard subsequence of this sequence.

18 is not in the sequence. It seems that, 18 is the only nonnegative integer which is not in the sequence.

References

• Paulo Ribenboim, The little book Of bigger primes, second edition, Springer, 2004, p. 185.

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..10000 (first 4230 terms from Hal M. Switkay)
• Carlos Rivera, Conjecture 30
• Alexei Kourbatov, Verification of the Firoozbakht conjecture for primes up to four quintillion, arXiv:1503.01744 [math.NT], 2015-2023.
• Alexei Kourbatov, Upper Bounds for Prime Gaps Related to Firoozbakht’s Conjecture, arXiv:1506.03042 [math.NT], 2015-2019.
• Alexei Kourbatov, Upper bounds for prime gaps related to Firoozbakht's conjecture, J. Int. Seq. 18 (2015) 15.11.2
• Wikipedia, Prime gap.
• Wikipedia, Firoozbakht Conjecture.

Crossrefs

Cf. A000040, A001223, A005669, A246777, A246778.

Cf. A249669.

Programs

Haskell

a246776 n = a249669 n - a000040 (n + 1)
-- Reinhard Zumkeller, Nov 16 2014

Mathematica

Table[Floor[Prime[n]^(1+1/n)]-Prime[n+1],{n,70}]

Formula

a(n) = A249669(n) - A000040(n+1). - Reinhard Zumkeller, Nov 16 2014