A327441 - OEIS

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

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OFFSET

2,2

COMMENTS

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

REFERENCES

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.

LINKS

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

EXAMPLE

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

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

MAPLE

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