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A248369 Least positive integer m such that prime(m+n) - prime(m) divides m.

%I A248369 #18 Jan 17 2022 11:16:39
%S A248369 1,6,54,18,26,24,80,120,180,48,70,160,92,82,220,98,228,102,378,130,
%T A248369 348,152,158,172,202,372,204,720,206,448,218,560,236,228,222,1480,282,
%U A248369 1656,636,300,312,322,764,350,356,352,362,420,434,860
%N A248369 Least positive integer m such that prime(m+n) - prime(m) divides m.
%C A248369 Conjecture: a(n) exists for any n > 0. Moreover, for n > 9 we have a(n) < n^2 except for n = 12, 19, 36, 38.
%C A248369 Note that for each n > 1 the term a(n) should be even and at least 2*n.
%H A248369 Zhi-Wei Sun, <a href="/A248369/b248369.txt">Table of n, a(n) for n = 1..10000</a>
%e A248369 a(7) = 80 since prime(80+7) - prime(80) = 449 - 409 = 40 divides 80.
%t A248369 Do[m=1;Label[aa];If[Mod[m,Prime[m+n]-Prime[m]]==0,Print[n," ",m];Goto[bb]];m=m+1;Goto[aa];Label[bb];Continue,{n,1,50}]
%t A248369 lpi[n_]:=Module[{m=1},While[Mod[m,Prime[n+m]-Prime[m]]!=0,m++];m]; Array[ lpi,50] (* _Harvey P. Dale_, Jan 17 2022 *)
%o A248369 (PARI)
%o A248369 a(n)=m=1;while(m%(prime(m+n)-prime(m)),m++);m
%o A248369 vector(100,n,a(n)) \\ _Derek Orr_, Oct 05 2014
%Y A248369 Cf. A000040, A247824, A248366.
%K A248369 nonn
%O A248369 1,2
%A A248369 _Zhi-Wei Sun_, Oct 05 2014