0,1

The first term corresponds to a gap of 1 = 2*(1/2) (so the offset might have been 1/2!).

a(n)=A000040(A038664(n)). - Lekraj Beedassy (blekraj(AT)yahoo.com), Sep 09 2006

Carella focuses on the problems of prime gaps and zero spacings. Possible solutions of several related problems such as the greatest lower bound, the least upper bound of the zero spacings, and the least upper bound of the prime gaps are considered. [From Jonathan Vos Post (jvospost3(AT)gmail.com), Mar 04 2009]

L. J. Lander and T. R. Parkin, On the first appearance of prime differences, Math. Comp., 21 (1967), 483-488.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

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

N. J. A. Sloane, Table of n, a(n) for n=0..603 (from the web page of Tomas Oliveira e Silva)

A. Booker, The Nth Prime Page

H. Bottomley, Prime number calculator

N. A. Carella, Note on Prime Gaps and Zero Spacing, Mar 3, 2009. [From Jonathan Vos Post (jvospost3(AT)gmail.com), Mar 04 2009]

T. R. Nicely, List of prime gaps

Tomas Oliveira e Silva, Gaps between consecutive primes

J. Thonnard, Les nombres premiers (Primality check; Closest next prime; Factorizer)

Index entries for primes, gaps between

The following table, based on a very much larger table in the web page of Tomas Oliveira e Silva (see link below) shows, for each gap g, P(g) = the smallest prime such that P(g)+g is the smallest prime number larger than P(g);

* marks a record-holder: g is a record-holder if P(g') > P(g) for all (even) g' > g, i.e. if all prime gaps are smaller than g for all primes smaller than P(g); P(g) is a record-holder if P(g') < P(g) for all (even) g' < g.

This table gives rise to many sequences: P(g) is A000230, the present sequence; P(g)* is A133430; the positions of the *'s in the P(g) column give A100180, A133430; g* is A005250; P(g*) is A002386; etc.

-----

g P(g)

-----

1* 2*

2* 3*

4* 7*

6* 23*

8* 89*

10 139*

12 199*

14* 113

16 1831*

18* 523

20* 887

22* 1129

24 1669

26 2477*

28 2971*

30 4297*

32 5591*

34* 1327

36* 9551*

........

The first time a gap of 4 occurs between primes is between 7 and 11, so a(2)=7 and A001632(2)=11.

a[n_] := If[n==1, 2, (For[m=1, Prime[m+1]-Prime[m]!= 2n-2, m++ ]; Prime[m])]; Table[a[n], {n, 50}] - from Farideh Firoozbakht (f.firoozbakht(AT)sci.ui.ac.irhea), Dec 17 2003

A001632(n) = 2n + a(n) = nextprime(a(n)).

Cf. A002386, A005250.

Cf. A100964 (least prime number that begins a prime gap of at least 2n).

For records see A133429, A133430, A100180.

Sequence in context: A129739 A163834 A002386 this_sequence A133429 A087770 A087164

Adjacent sequences: A000227 A000228 A000229 this_sequence A000231 A000232 A000233

nonn,nice

N. J. A. Sloane (njas(AT)research.att.com).

More terms from Jud McCranie j.mccranie(AT)comcast.net. Further terms from Robert A. Stump (bee_ess107(AT)yahoo.com), Jan 11 2002