A246777 a(n) = A246776(A005669(n)): using the indices of maximal primes in A002386 in order to verify the Firoozbakht conjecture for 0 <= floor(prime(n)^(1+1/n)) - prime(n+1).

1, 0, 0, 3, 10, 5, 16, 19, 20, 10, 38, 38, 35, 24, 43, 53, 38, 43, 66, 52, 46, 65, 79, 55, 73, 104, 109, 95, 120, 92, 130, 130, 121, 127, 114, 127, 155, 148, 92, 109, 159, 171, 173, 180, 171, 157, 171, 161, 174, 178, 168, 165, 169, 135, 171, 168, 138, 174, 195, 234, 149, 253, 269, 61, 244, 248, 255, 323, 304, 307, 262, 245, 234, 215, 228

Offset

1,4

Comments

a(1) > 0 and a(n) >= 0 for n < 76; this implies "if p=p(k) is in the sequence A002386 and p <= 1425172824437699411 then p(k+1)^(1/(k+1)) < p(k)^(1/k)."

Links

Table of n, a(n) for n=1..75.

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

A. Kourbatov, Upper bounds for prime gaps related to Firoozbakht's conjecture, J. Int. Seq. 18 (2015) 15.11.2

Wikipedia, Firoozbakht's Conjecture

Formula

a(n) = A246776(A005669(n)).

Mathematica

f[n_] := Block[{d, i, j, m = 0}, Reap@ For[i = 1, i <= n, i++, d = Prime[i + 1] - Prime@ i; If[d > m, m = d; Sow@ i, False]] // Flatten // Rest] (* A005669 *); g[n_] := Floor[Prime[n]^(1 + 1/n)] - Prime[n + 1] (* A246776 *); g@ f@ 100000; (* Michael De Vlieger, Mar 24 2015, with code from A246776 by Farideh Firoozbakht *)

Crossrefs

Cf. A000040, A005250, A005669, A002386, A246776, A246778, A246779, A246789.

Sequence in context: A035411 A033478 A285576 * A111127 A140948 A328763

Adjacent sequences: A246774 A246775 A246776 * A246778 A246779 A246780

Keyword

nonn

Author

Farideh Firoozbakht, Sep 30 2014

Status

approved