A257933 Prime p such that sqrt(p+2) is semiprime (A001358).

79, 223, 439, 1087, 1223, 2399, 3023, 4759, 5927, 8647, 14159, 14639, 21023, 24023, 25919, 28559, 31327, 33487, 42023, 47087, 56167, 61007, 64007, 67079, 70223, 71287, 89399, 90599, 91807, 95479, 104327, 112223, 116279, 126023, 137639, 152879, 172223, 199807

Offset: 1

Author

Vladimir Shevelev, May 13 2015

Keywords

• nonn

Comments

The terms are not congruent to 1 (mod 10).

The sequence contains no Mersenne prime p=2^t-1. Since p > 79, t is an odd prime and p+2 = 2^t+1 is divisible by 3. So, since 2^t+1 should be square, 2^t+1 is divisible by 9, i.e., (2^t+1)/3 == 0 (mod 3). (1)

Note that either t=6k+1 or t=6m+5. In each case, (1) is impossible.

Indeed, if t=6k+1, then (2^t+1)/3 = (2*(4^k)^3+1)/3 = (2*(3+1)^(3*k)+1)/3 == (2*binomial(3*k,1)*3+2+1)/3 == 1(mod 3), and analogously in case t=6*m+5, (2^t+1)/3 == 2 (mod 3): a contradiction.

Examples

			Prime 79 is in the sequence because sqrt(79+2) = 9 = 3*3 which is semiprime.
Prime 1223 is in the sequence because sqrt(1223+2) = 35 = 5*7 which is semiprime.
		

Links

• Peter J. C. Moses and Charles R Greathouse IV, Table of n, a(n) for n = 1..10000 (first 1000 terms from Moses)

Crossrefs

Cf. A000040, A001358, A046315.

Programs

Mathematica

Select[Prime@Range@18000,PrimeOmega[Sqrt[#+2]]==2&]//Quiet (* Ivan N. Ianakiev, May 13 2015 *)

PARI

issemi(n)=bigomega(n)==2
is(n)=isprime(n) && issquare(n+2,&n) && issemi(n) \\ Charles R Greathouse IV, May 13 2015

PARI

list(lim)=my(v=List(), k=sqrt(lim+2), t); forprime(p=2, sqrt(k), forprime(q=p, k\p, if(isprime(t=(p*q)^2-2), listput(v, t)))); Set(v) \\ Charles R Greathouse IV, May 13 2015

Perl

use ntheory ":all"; forprimes { say if is_power($+2,2) && scalar(factor(sqrtint($+2)))==2 } 1e7; # Dana Jacobsen, May 13 2015

Perl

use ntheory ":all"; sub list { my($lim,$k,$t,$p,%v)=shift; $k=sqrt($lim+2); forprimes { $p=$; forprimes { $t=($p*$)**2-2; $v{$t}++ if is_prime($t); } $p,int($k/$p); } int(sqrt($k)); my @v=sort{$a<=>$b} keys %v; @v; } say for list(1e10); # Translation of PARI, Dana Jacobsen, May 13 2015

Formula

Trivially a(n) >> n^2 log^2 n/(log log n)^2. - Charles R Greathouse IV, May 13 2015

Extensions

More terms from Peter J. C. Moses, May 13 2015