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 A145823 Squares of the form p1 - 1 where p1 is a lower twin prime. 0

%I

%S 4,16,100,196,5476,8836,16900,17956,21316,25600,52900,65536,106276,

%T 115600,122500,164836,184900,193600,220900,341056,401956,470596,

%U 490000,495616,614656,739600,846400,972196,1110916,1144900,1336336,1464100

%N Squares of the form p1 - 1 where p1 is a lower twin prime.

%C These numbers are of the form 3n+1. This follows from the fact that a lower twin prime > 3 is of the form 3n+2. If p1 = 3n+1 then the upper twin would be 3n+1+2 = 3k which is not prime for k > 1. 4 is the only square which is a lower twin prime + 1. If a lower twin prime p1 + 1 is a square, then it is of the form 3n+2+1 or 3n. Then 3n = x^2 implies x = 3r for some r. This implies 3n = 9r^2. Now if we subtract 1 we have 9r^2-1 = (3r-1)(3r+1) which is not prime.

%F a(n) = A080149(n)^2. [From _Ray Chandler_, Oct 24 2008]

%e p1=5 is a lower twin prime. 5-1 = 4 is a square.

%t Select[Select[Partition[Prime[Range[112000]],2,1],#[[2]]-#[[1]]==2&][[All,1]]-1,IntegerQ[Sqrt[#]]&] (* _Harvey P. Dale_, Jul 05 2020 *)

%o (PARI) g(n) = for(x=1,n,y=twinl(x)-1;if(issquare(y),print1(y",")))

%o twinl(n) = local(c, x); c=0;x=1;while(c<n,if(ispseudoprime(prime(x)+2),c++);

%o x++;);return(prime(x-1))

%K nonn

%O 1,1

%A _Cino Hilliard_, Oct 20 2008

%E Extended by _Ray Chandler_, Oct 24 2008

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Last modified May 27 03:13 EDT 2022. Contains 354093 sequences. (Running on oeis4.)