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

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

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

Table of n, a(n) for n=1..32.

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[112000]], 2, 1], #[[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<n, if(ispseudoprime(prime(x)+2), c++);
x++; ); return(prime(x-1))

Crossrefs

Sequence in context: A091040 A029985 A114023 * A121142 A318693 A087296

Adjacent sequences: A145820 A145821 A145822 * A145824 A145825 A145826

Keyword

nonn

Author

Cino Hilliard, Oct 20 2008

Extensions

Extended by Ray Chandler, Oct 24 2008

Status

approved