A091305 Primes of the form p*q - p - q, where p and q are primes.

3, 5, 7, 11, 17, 19, 23, 29, 31, 41, 43, 47, 59, 71, 79, 83, 101, 103, 107, 131, 137, 139, 149, 163, 167, 179, 191, 197, 199, 211, 223, 227, 239, 251, 263, 269, 271, 281, 311, 331, 347, 359, 379, 383, 419, 431, 443, 461, 463, 467, 479, 491, 499, 503, 521, 523

Offset: 1

Author

Zak Seidov, Feb 21 2004

Keywords

• easy
• nice
• nonn

Comments

Some primes have more than one representation (besides of symmetry in p,q!), e.g. 11 with (p,q)=(2,13) and (3,7).

If (r,r+2) is a twin prime pair then r is in this sequence (with q=2, p=r+2). - Emmanuel Vantieghem, Jun 02 2025

Examples

			31 is a member with p=3, q=17.
		

Links

• T. D. Noe, Table of n, a(n) for n=1..1000

Crossrefs

Cf. A066938 (p*q + p + q), A091301 (p*q + p - q).

Cf. A001359 (subsequence).

Programs

Maple

N:= 1000: # for terms <= N
P:= select(isprime, [2,seq(i,i=3..N/2,2)]):
S:= {}:
for i from 1 to nops(P) do
  for j from 1 to i do
    x:= P[i]*P[j]-P[i]-P[j];
    if x > N then break fi;
    if isprime(x) then S:= S union {x} fi
od od:
sort(convert(S,list)); # Robert Israel, Jun 05 2025

Mathematica

mp[{p_,q_}]:=p*q-p-q; Take[Union[Select[mp/@Subsets[Prime[Range[100]],{2}], PrimeQ]],60] (* Harvey P. Dale, Nov 27 2011 *)

PARI

isA091305(p)=fordiv(p++,d,if(isprime(d+1)&isprime(p/d+1), return(isprime(p-1)))) \\ Charles R Greathouse IV, Feb 15 2011