A005669 Indices of primes where largest gap occurs.

1, 2, 4, 9, 24, 30, 99, 154, 189, 217, 1183, 1831, 2225, 3385, 14357, 30802, 31545, 40933, 103520, 104071, 149689, 325852, 1094421, 1319945, 2850174, 6957876, 10539432, 10655462, 20684332, 23163298, 64955634, 72507380, 112228683, 182837804, 203615628, 486570087

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn

Comments

Conjecture: log a(n) ~ n/2. That is, record prime gaps occur about twice as often as records in an i.i.d. random sequence of comparable length (see arXiv:1709.05508 for a heuristic explanation). - Alexei Kourbatov, Mar 28 2018

References

• H. Riesel, Prime numbers and computer methods for factorization, Progress in Mathematics, Vol. 57, Birkhäuser, Boston, 1985, Chap. 4, see pp. 381-384.
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• John W. Nicholson, Table of n, a(n) for n = 1..82 (first 77 terms from Charles R Greathouse IV)
• Jens Kruse Andersen, The Top-20 Prime Gaps.
• Jens Kruse Andersen, New record prime gap.
• Jens Kruse Andersen, Maximal gaps.
• R. K. Guy, Letter to N. J. A. Sloane, 1987.
• Alexei Kourbatov, Tables of record gaps between prime constellations, arXiv preprint arXiv:1309.4053 [math.NT], 2013.
• Alexei Kourbatov, On the nth record gap between primes in an arithmetic progression, arXiv:1709.05508 [math.NT], 2017; Int. Math. Forum, 13 (2018), 65-78.
• Alexei Kourbatov and Marek Wolf, Predicting maximal gaps in sets of primes, arXiv preprint arXiv:1901.03785 [math.NT], 2019.
• Thomas R. Nicely, First occurrence prime gaps. [For local copy see A000101]
• Matt Visser, Verifying the Firoozbakht, Nicholson, and Farhadian conjectures up to the 81st maximal prime gap, arXiv:1904.00499 [math.NT], 2019.
• Robert G. Wilson v, Notes (no date).
• J. Young and A. Potler, First occurrence prime gaps, Math. Comp., 52 (1989), 221-224.
• Index entries for primes, gaps between

Crossrefs

Cf. A000101, A002386, A005250.

Programs

Mathematica

f[n_] := Block[{d, i, 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]; f@ 1000000 (* Michael De Vlieger, Mar 24 2015 *)

Formula

a(n) = A000720(A002386(n)).

a(n) = A107578(n) - 1. - Jens Kruse Andersen, Oct 19 2010