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

1,1

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

Links

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

Example

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

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 *)

Prog

(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

Crossrefs

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

Cf. A001359 (subsequence).

Sequence in context: A082373 A116959 A249505 * A164319 A085498 A225223

Adjacent sequences: A091302 A091303 A091304 * A091306 A091307 A091308

Keyword

easy,nice,nonn

Author

Zak Seidov, Feb 21 2004

Status

approved