A053695 Differences between record prime gaps.

1, 2, 2, 2, 6, 4, 2, 2, 12, 2, 8, 8, 20, 14, 10, 16, 2, 4, 14, 16, 6, 26, 30, 10, 2, 12, 14, 2, 32, 6, 4, 28, 16, 18, 28, 2, 10, 62, 8, 4, 6, 12, 4, 10, 14, 2, 16, 2, 6, 42, 6, 14, 50, 22, 42, 50, 12, 26, 2, 100, 10, 8, 208, 52, 14, 22, 4, 24, 24, 56, 28, 14, 72, 34, 12, 22

Offset: 1

Author

Jeff Burch, Mar 23 2000

Keywords

• nonn
• nice
• hard

Comments

The largest known term of this sequence is a(63) = 1132 - 924 = 208. This seems rather strange for a(63) > 2*100+7 where 100 = max {a(k)| k < 63}. {1,2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,32,34,42,50,52,56,62,72,100,208} is the set of the distinct first 75 terms of the sequence. What is the smallest number m such that a(m) = 36? - Farideh Firoozbakht, May 30 2014

Conjecture: a(n) <= A005250(n). Based on the equivalent statement at A005250: A005250(n+1) / A005250(n) <= 2. - John W. Nicholson, Dec 30 2015

Links

• John W. Nicholson, Table of n, a(n) for n = 1..81
• Alexei Kourbatov, On the nth record gap between primes in an arithmetic progression. International Mathematical Forum, Vol. 13, 2018, No. 2, 65-78, arXiv:1709.05508v3 [math.NT].
• Eric Weisstein's World of Mathematics, Prime Gaps
• Wikipedia, Prime gap

Crossrefs

Cf. A000101, A001223, A002386, A005250, A053686.

Programs

Mathematica

m = 2; r = 0; Differences@ Reap[Monitor[Do[If[Set[d, Set[n, NextPrime[m]] - m] > r, Set[r, d]; Sow[d]]; m = n, {i, 10^7}], i]][[-1, -1]] (* Michael De Vlieger, Oct 30 2021 *)

Formula

a(n) = A005250(n+1) - A005250(n).

A005250(n+1) = 1 + Sum_{i=1..n} a(i). - John W. Nicholson, Dec 29 2015

Extensions

Missing term 1 and more terms added by Farideh Firoozbakht, May 30 2014

a(75)-a(76) from John W. Nicholson, Feb 27 2018