A005669 Indices of primes where largest gap occurs. (Formerly M1193)

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

1,2

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.

Vincent Granville, Piercing Gilbreath's Conjecture: From Deep Number Theory Insights to Fintech and Cybersecurity, arXiv:2607.04166 [cs.CR], 2026. See p. 20 (Table 19).

Richard 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).

Jeff Young and Aaron Potler, First occurrence prime gaps, Math. Comp. 52 (1989), 221-224.

Index entries for primes, gaps between

Formula

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

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

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 *)

Crossrefs

Cf. A000101, A002386, A005250.

Sequence in context: A229048 A144309 A080376 * A038664 A261367 A148077

Adjacent sequences: A005666 A005667 A005668 * A005670 A005671 A005672

Keyword

nonn

Author

N. J. A. Sloane

Status

approved