%I #86 Aug 04 2024 12:20:40
%S 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,
%T 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,
%U 42,50,12,26,2,100,10,8,208,52,14,22,4,24,24,56,28,14,72,34,12,22
%N Differences between record prime gaps.
%C 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
%C Conjecture: a(n) <= A005250(n). Based on the equivalent statement at A005250: A005250(n+1) / A005250(n) <= 2. - _John W. Nicholson_, Dec 30 2015
%H John W. Nicholson, <a href="/A053695/b053695.txt">Table of n, a(n) for n = 1..81</a>
%H Alexei Kourbatov, <a href="https://arxiv.org/abs/1709.05508v3">On the nth record gap between primes in an arithmetic progression.</a> International Mathematical Forum, Vol. 13, 2018, No. 2, 65-78, arXiv:1709.05508v3 [math.NT].
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PrimeGaps.html">Prime Gaps</a>
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Prime_gap">Prime gap</a>
%F a(n) = A005250(n+1) - A005250(n).
%F A005250(n+1) = 1 + Sum_{i=1..n} a(i). - _John W. Nicholson_, Dec 29 2015
%t 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 *)
%Y Cf. A000101, A001223, A002386, A005250, A053686.
%K nonn,nice,hard
%O 1,2
%A _Jeff Burch_, Mar 23 2000
%E Missing term 1 and more terms added by _Farideh Firoozbakht_, May 30 2014
%E a(75)-a(76) from _John W. Nicholson_, Feb 27 2018