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

1,5

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

A. Kourbatov, Verification of the Firoozbakht conjecture for primes up to four quintillion, arXiv:1503.01744 [math.NT], 2015

Alexei Kourbatov, Upper Bounds for Prime Gaps Related to Firoozbakht’s Conjecture, arXiv preprint, 2015.

A. 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.

Formula

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

Mathematica

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

Prog

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

Crossrefs

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

Cf. A249669.

Sequence in context: A272101 A059853 A136527 * A138614 A161912 A162339

Adjacent sequences: A246773 A246774 A246775 * A246777 A246778 A246779

Keyword

nonn

Author

Farideh Firoozbakht, Sep 26 2014

Status

approved