

A053695


Differences between record prime gaps.


7



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
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OFFSET

1,2


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..79
Alexei Kourbatov, On the nth record gap between primes in an arithmetic progression. International Mathematical Forum, Vol. 13, 2018, No. 2, 6578, arXiv:1709.05508v3 [math.NT].
Eric Weisstein's World of Mathematics, Prime Gaps
Wikipedia, Prime gap


FORMULA

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


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 *)


CROSSREFS

Cf. A000101, A001223, A002386, A005250, A053686.
Sequence in context: A126615 A158524 A054274 * A210550 A208659 A209752
Adjacent sequences: A053692 A053693 A053694 * A053696 A053697 A053698


KEYWORD

nonn,nice,hard


AUTHOR

Jeff Burch, Mar 23 2000


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


STATUS

approved



