A166594 -id:A166594 - cp's OEIS frontend

A327441 a(n) = max_{p <= n} (p'-p), where p and p' are successive primes.

1, 2, 2, 2, 2, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 14, 14, 14, 14, 14, 14, 14

Offset: 2

N. J. A. Sloane, Sep 11 2019

nonn

This is Maier and Pomerance's G(n).

			a(2) = 1 from p=2, p'=3.
a(3) = 2 from p=3, p'=5.

Erdos, Paul. "On the difference of consecutive primes." The Quarterly Journal of Mathematics 1 (1935): 124-128.
Erdös, P. "On the difference of consecutive primes." Bulletin of the American Mathematical Society 54.10 (1948): 885-889.
Maier, Helmut, and Carl Pomerance. "Unusually large gaps between consecutive primes." Transactions of the American Mathematical Society 322.1 (1990): 201-237.
D. S. Mitrinovic et al., Handbook of Number Theory, Kluwer, 1996, Section VII.22, p. 249. (See G(x). Gives bounds.)
Rankin, Robert Alexander. "The difference between consecutive prime numbers V." Proceedings of the Edinburgh Mathematical Society 13.4 (1963): 331-332.

N. J. A. Sloane, Table of n, a(n) for n = 2..20000

Cf. A063095.

A166594 is a similar sequence, but the present sequence matches the definition used by Maier and Pomerance.

with(numtheory);
M:=120; a:=[]; r:=0;
for x from 2 to M do
  i1:=pi(x); p:=ithprime(i1); q:=ithprime(i1+1); d:=q-p;
    if d>r then r:=d; fi;
a:=[op(a),r]; od: a; # N. J. A. Sloane, Sep 11 2019

A166597 Let p = largest prime <= n, with p(0)=p(1)=0, and let q = smallest prime > n; then a(n) = q-p.

Original entry on oeis.org

2, 2, 1, 2, 2, 2, 2, 4, 4, 4, 4, 2, 2, 4, 4, 4, 4, 2, 2, 4, 4, 4, 4, 6, 6, 6, 6, 6, 6, 2, 2, 6, 6, 6, 6, 6, 6, 4, 4, 4, 4, 2, 2, 4, 4, 4, 4, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 2, 2, 6, 6, 6, 6, 6, 6, 4, 4, 4, 4, 2, 2, 6, 6, 6, 6, 6, 6, 4, 4, 4, 4, 6, 6, 6, 6, 6, 6, 8, 8, 8, 8, 8, 8, 8, 8, 4, 4, 4, 4, 2, 2, 4, 4

Offset: 0

Daniel Forgues, Oct 17 2009

nonn

Note the large prime gap of 72 between 31397 and 31469. This is the prime gap with the largest merit (cf. A111870), 72/log(31397)=6.95352 for primes less than 100000. Also 72/(log(31397))^2=0.67154 (cf. conjectures of Cramer-Granville, Shanks and Wolf) is largest for primes less than 100000. - Daniel Forgues, Oct 23 2009

			a(0) = 2 since the least prime greater than 0 is 2 (gap of 2 from 0 to 2).
a(9) = 4 since the least prime greater than 9 is 11 (gap of 4 from 7 to 11).
a(11) = 2 since the least prime greater than 11 is 13 (gap of 2 from 11 to 13).

Daniel Forgues, Table of n, a(n) for n = 0..100000
Eric Weisstein's World of Mathematics, Prime Gaps.
Eric Weisstein's World of Mathematics, Cramer-Granville Conjecture.
Eric Weisstein's World of Mathematics, Shanks Conjecture.

Cf. A151800, A166594.
Cf. A111870. - Daniel Forgues, Oct 23 2009
See A327441 for the classic G(n) version. - N. J. A. Sloane, Sep 11 2019
Cf. A001223, A000720, A007917, A007918, A151799.

2,2,seq(nextprime(n)-prevprime(n+1), n=2..100); # Ridouane Oudra, Dec 28 2024

f[n_]:=Module[{a=If[PrimeQ[n],n,NextPrime[n,-1]]}, NextPrime[n]-a]; Join[{2,2},Array[f,120,2]] (* Harvey P. Dale, May 17 2011 *)

a(n) = nextprime(n+1) - precprime(n); \\ Michel Marcus, Mar 02 2023

From Ridouane Oudra, Dec 28 2024: (Start)
a(n) = A001223(A000720(n)), for n>1.
a(n) = A151800(n) - A007917(n), for n>1.
a(n) = A007918(n+1) - A151799(n+1), for n>1. (End)

Definition rephrased by N. J. A. Sloane, Oct 25 2009

cp's OEIS Frontend

A327441 a(n) = max_{p <= n} (p'-p), where p and p' are successive primes.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Maple