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

1,1

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.

Links

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

Formula

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

Example

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.

Mathematica

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

Prog

(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

Crossrefs

Cf. A000040, A001358, A046315.

Sequence in context: A089686 A278837 A260335 * A258098 A141964 A347032

Adjacent sequences: A257930 A257931 A257932 * A257934 A257935 A257936

Keyword

nonn

Author

Vladimir Shevelev, May 13 2015

Extensions

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

Status

approved