|
| |
|
|
A145824
|
|
Lower twin primes p1 such that p1-1 is a square.
|
|
2
| |
|
|
5, 17, 101, 197, 5477, 8837, 16901, 17957, 21317, 25601, 52901, 65537, 106277, 115601, 122501, 164837, 184901, 193601, 220901, 341057, 401957, 470597, 490001, 495617, 614657, 739601, 846401, 972197, 1110917, 1144901, 1336337, 1464101
(list; graph; refs; listen; history; internal format)
|
|
|
|
OFFSET
| 1,1
|
|
|
COMMENTS
| 3 is the only lower twin prime for which p1+1 is a square. This follows from
of the fact that lower twin primes > 3 are of the form 3n+2 and adding 1 we get
a number of the form 3m. Then 3m = k^2 implies k = 3r and 3m = 9r^2.
Subtracting 1 we have 3m = (3r-1)(3r+1) not prime contradicting 3m-1 is prime.
Conjecture: Primes of this form are infinite.
a(n)= A080149(n)^2 +1. [From Zak Seidov (zakseidov(AT)yahoo.com), Oct 21 2008]
|
|
|
LINKS
| Zak Seidov, Table of n, a(n) for n=1..4663, a(n)<10^12
|
|
|
EXAMPLE
| p1=5 is a lower twin prime. 5-1 = 4 is a square.
|
|
|
MATHEMATICA
| lst={}; Do[p=Prime@n; If[PrimeQ@(p+2)&&Sqrt[p-1]==IntegerPart[Sqrt[p-1]], AppendTo[lst, p]], {n, 9!}]; lst [From Vladimir Orlovsky, Aug 11 2009]
|
|
|
PROG
| (PARI) g(n) = for(x=1, n, y=twinl(x)-1; if(issquare(y), print1(y+1", ")))
twinl(n) = local(c, x); c=0; x=1; while(c<n, if(ispseudoprime(prime(x)+2), c++);
x++; ); return(prime(x-1))
|
|
|
CROSSREFS
| A080149 [From Zak Seidov (zakseidov(AT)yahoo.com), Oct 21 2008]
Subsequence of A002496 (Primes of form n^2 + 1). - Moshe Levin, Nov 25 2011
Sequence in context: A198090 A180382 A139390 * A076516 A145986 A200992
Adjacent sequences: A145821 A145822 A145823 * A145825 A145826 A145827
|
|
|
KEYWORD
| nonn
|
|
|
AUTHOR
| Cino Hilliard (hillcino368(AT)hotmail.com), Oct 20 2008
|
|
|
EXTENSIONS
| More terms from Zak Seidov (zakseidov(AT)yahoo.com), Oct 21 2008
|
| |
|
|
