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 A145823 Squares of the form p1 - 1 where p1 is a lower twin prime. 0
 4, 16, 100, 196, 5476, 8836, 16900, 17956, 21316, 25600, 52900, 65536, 106276, 115600, 122500, 164836, 184900, 193600, 220900, 341056, 401956, 470596, 490000, 495616, 614656, 739600, 846400, 972196, 1110916, 1144900, 1336336, 1464100 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS 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. LINKS FORMULA a(n) = A080149(n)^2. [From Ray Chandler, Oct 24 2008] EXAMPLE p1=5 is a lower twin prime. 5-1 = 4 is a square. MATHEMATICA Select[Select[Partition[Prime[Range], 2, 1], #[]-#[]==2&][[All, 1]]-1, IntegerQ[Sqrt[#]]&] (* Harvey P. Dale, Jul 05 2020 *) PROG (PARI) g(n) = for(x=1, n, y=twinl(x)-1; if(issquare(y), print1(y", "))) twinl(n) = local(c, x); c=0; x=1; while(c

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Last modified January 23 14:34 EST 2022. Contains 350512 sequences. (Running on oeis4.)