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

1,1

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

References

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

Links

Table of n, a(n) for n=1..55.

Formula

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

Example

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.

Prog

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

Crossrefs

Cf. A000040, A001223, A005250, A005669.

Cf. A053695, A144309.

Sequence in context: A340714 A292381 A233655 * A272196 A335057 A039887

Adjacent sequences: A307322 A307323 A307324 * A307326 A307327 A307328

Keyword

nonn

Author

Marius A. Burtea, Apr 02 2019

Status

approved