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A213649 Smallest k such that there exists a square between prime(n) and prime(n+k). 1

%I

%S 2,1,2,1,2,1,3,2,1,2,1,4,3,2,1,3,2,1,4,3,2,1,3,2,1,5,4,3,2,1,4,3,2,1,

%T 5,4,3,2,1,5,4,3,2,1,4,3,2,1,6,5,4,3,2,1,7,6,5,4,3,2,1,5,4,3,2,1,6,5,

%U 4,3,2,1,6,5,4,3,2,1,7,6,5,4,3,2,1,7,6

%N Smallest k such that there exists a square between prime(n) and prime(n+k).

%C a(A038107(n)) = 1 for n >= 2.

%C a(n) is of the form {S1} union {S2} union ... union {Sk} union ... where a subset Sk is of the form {xk, xk - 1, xk - 2, …, 1 }. We obtain a subsequence Max {Sn} = {xn} = {2, 2, 2, 3, 2, 4, 3, 4, 3, 5, 4, 5, 5, 4, 6, 7, 5, 6, 6, 7, 7, 7, 6, 9, …}.

%H Michel Lagneau, <a href="/A213649/b213649.txt">Table of n, a(n) for n = 1..5000</a>

%e a(7)=3 because prime(7) = 17, prime(7+3) = 29 and 17 < 25 < 29 where 25 is square.

%p with(numtheory):for n from 1 to 100 do:ii:=0:for k from 1 to 100 while(ii=0) do:p1:=ithprime(n):p2:=ithprime(n+k):i:=0:for m from p1+1 to p2-1 do:c:=sqrt(m):if c=floor(c) then i:=i+1:else fi:od: if i<>0 then ii:=1:printf(`%d, `,k):else fi:od:od:

%Y Cf. A038107, A014085.

%K nonn

%O 1,1

%A _Michel Lagneau_, Jun 17 2012

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Last modified April 19 20:08 EDT 2021. Contains 343117 sequences. (Running on oeis4.)