A307325 a(n) is the smallest number k for which prime(k+1) - prime(k) is greater than n.

2, 4, 4, 9, 9, 24, 24, 30, 30, 30, 30, 30, 30, 99, 99, 99, 99, 154, 154, 189, 189, 217, 217, 217, 217, 217, 217, 217, 217, 217, 217, 217, 217, 1183, 1183, 1831, 1831, 1831, 1831, 1831, 1831, 1831, 1831, 2225, 2225, 2225, 2225, 2225, 2225, 2225, 2225, 3385, 3385, 3385, 3385

Offset: 1

Author

Marius A. Burtea, Apr 02 2019

Keywords

• nonn

Comments

For any n there is an infinity of numbers m for which prime(m+1) - prime(m) is greater than n.

It appears that the sequence of lengths of successive runs is equal to A053695. - Marc Bofill Janer, May 21 2019

Examples

			For n = 2, prime(2) - prime(1) = 3 - 2 = 1, prime(3) - prime(2) = 5 - 3 = 2, prime(5) - prime(4) = 11 - 7 = 4, so a(2) = 4.
		

References

• Laurențiu Panaitopol, Dinu Șerbănescu, Number theory and combinatorial problems for juniors, Ed.Gil, Zalău, (2003), ch. 1, p.7, pr. 25. (in Romanian).

Crossrefs

Cf. A000040, A001223, A005250, A005669.

Cf. A053695, A144309.

Programs

MATLAB

v=primes(1000000);
for u=1:100; ss=1;
    while and(v(ss+1)-v(ss)<=u,ss
				

Magma

v:=PrimesUpTo(10000000);
sol:=[];
for u in [1..60] do
   for ss in [1..#v-1] do
    if v[ss+1]-v[ss] gt u then
         sol[u]:=ss;
         break;
     end if;
   end for;
end for;
   sol;

PARI

a(n) = my(k=1); while(prime(k+1) - prime(k) <= n, k++); k; \\ Michel Marcus, Apr 03 2019

Formula

a(2*n) = a(2*n+1) = A144309(n+1) for n>=1. - Georg Fischer, Dec 05 2022