Journal of Integer Sequences, Vol. 23 (2020), Article 20.9.3 |

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Redmond, WA 98052

USA

Marek Wolf

Faculty of Mathematics and Natural Sciences

Cardinal Stefan Wyszyński University

Warsaw, PL-01-938

Poland

**Abstract:**

We study the first occurrences of gaps between primes in the arithmetic progression
(P): *r*, *r* + *q*,
*r* + 2*q*,
*r* + 3*q*, . . . ,
where *q* and *r* are coprime integers, *q* > *r* ≥ 1.
The growth trend and distribution of the first-occurrence gap sizes are similar to those
of maximal gaps between primes in (P). The histograms of first-occurrence gap sizes,
after appropriate rescaling, are well approximated by the Gumbel extreme value distribution.
Computations suggest that first-occurrence gaps are much more numerous than maximal gaps:
there are *O*(log^{2 }*x*) first-occurrence gaps between primes in (P)
below *x*, while the number of maximal gaps is only *O*(log *x*). We explore
the connection between the asymptotic density of gaps of a given size and the corresponding
generalization of Brun's constant. For the first occurrence of gap *d* in (P), we expect
the end-of-gap prime
*p* ≍
√*d*
exp√*d/φ*(*q*)
infinitely often. Finally, we study the gap size as a function of its index
in the sequence of first-occurrence gaps.

(Concerned with sequences A000010 A001620 A005597 A014320 A065421 A167864 A194098 A268925 A268928 A330853 A330854 A330855 A334543 A334544 A334545 A335366 A335367.)

Received June 11 2020; revised versions received September 15 2020,
September 19 2020.
Published in *Journal of Integer Sequences*,
October 15 2020.

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