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A378111 a(n) is the least prime p such that there are exactly n squarefree numbers strictly between p and the next prime, or -1 if there is no such p.

%I A378111 #35 Dec 01 2024 17:57:02
%S A378111 2,5,13,31,89,139,113,199,211,317,1759,1381,1951,887,4523,2179,2477,
%T A378111 4831,5351,4297,1327,9973,14107,19333,16141,20809,15683,37907,28229,
%U A378111 58831,31907,19609,25471,40289,114493,43331,44293,34061,191353,31397,107377,134513,186481,448451,175141,332317,188029
%N A378111 a(n) is the least prime p such that there are exactly n squarefree numbers strictly between p and the next prime, or -1 if there is no such p.
%C A378111 a(n) = A000040(k) where k is the least number such that A061398(k) = n.
%H A378111 David A. Corneth, <a href="/A378111/b378111.txt">Table of n, a(n) for n = 0..253</a> (first 144 terms from Robert Israel)
%H A378111 David A. Corneth, <a href="/A378111/a378111.gp.txt">PARI program</a>
%e A378111 a(3) = 31 because there are 3 squarefree numbers between 31 and the next prime 37, namely 33, 34 and 35, and 31 is the least prime that works.
%p A378111 V:= Array(0..100): count:= 0: q:= 2:
%p A378111 for k from 1 while count < 101 do
%p A378111   p:= q; q:= nextprime(q);
%p A378111   v:= nops(select(numtheory:-issqrfree,[$p+1 .. q-1]));
%p A378111   if v <= 100 and V[v] = 0 then
%p A378111     V[v]:= p; count:= count+1;
%p A378111   fi
%p A378111 od:
%Y A378111 Cf. A000040, A005117, A061398, A377430.
%K A378111 nonn
%O A378111 0,1
%A A378111 _Robert Israel_, Nov 29 2024