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

%I A307325 #22 Dec 06 2022 02:52:00
%S A307325 2,4,4,9,9,24,24,30,30,30,30,30,30,99,99,99,99,154,154,189,189,217,
%T A307325 217,217,217,217,217,217,217,217,217,217,217,1183,1183,1831,1831,1831,
%U A307325 1831,1831,1831,1831,1831,2225,2225,2225,2225,2225,2225,2225,2225,3385,3385,3385,3385
%N A307325 a(n) is the smallest number k for which prime(k+1) - prime(k) is greater than n.
%C A307325 For any n there is an infinity of numbers m for which prime(m+1) - prime(m) is greater than n.
%C A307325 It appears that the sequence of lengths of successive runs is equal to A053695. - _Marc Bofill Janer_, May 21 2019
%D A307325 Laurențiu Panaitopol, Dinu Șerbănescu, Number theory and combinatorial problems for juniors, Ed.Gil, Zalău, (2003), ch. 1, p.7, pr. 25. (in Romanian).
%F A307325 a(2*n) = a(2*n+1) = A144309(n+1) for n>=1. - _Georg Fischer_, Dec 05 2022
%e A307325 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.
%o A307325 (MATLAB) v=primes(1000000);
%o A307325 for u=1:100; ss=1;
%o A307325     while and(v(ss+1)-v(ss)<=u,ss<length(v)-1); ss=ss+1;end;
%o A307325     sol(u)=ss;
%o A307325 end
%o A307325    sol
%o A307325 (Magma) v:=PrimesUpTo(10000000);
%o A307325 sol:=[];
%o A307325 for u in [1..60] do
%o A307325    for ss in [1..#v-1] do
%o A307325     if v[ss+1]-v[ss] gt u then
%o A307325          sol[u]:=ss;
%o A307325          break;
%o A307325      end if;
%o A307325    end for;
%o A307325 end for;
%o A307325    sol;
%o A307325 (PARI) a(n) = my(k=1); while(prime(k+1) - prime(k) <= n, k++); k; \\ _Michel Marcus_, Apr 03 2019
%Y A307325 Cf. A000040, A001223, A005250, A005669.
%Y A307325 Cf. A053695, A144309.
%K A307325 nonn
%O A307325 1,1
%A A307325 _Marius A. Burtea_, Apr 02 2019