A330477 Semiprimes (A001358) p*q such that p*q+p+q is also a semiprime.

9, 22, 25, 39, 62, 69, 77, 87, 91, 94, 95, 106, 115, 119, 121, 122, 133, 134, 142, 146, 159, 183, 187, 202, 213, 214, 218, 219, 226, 235, 237, 249, 253, 259, 262, 265, 274, 287, 289, 291, 299, 303, 305, 309, 314, 335, 362, 381, 386, 393, 403, 411, 417, 422, 446, 458, 469, 473, 489, 501, 502, 505

Offset: 1

Author

J. M. Bergot and Robert Israel, Dec 15 2019

Keywords

• nonn

Examples

			a(3) = 25 is a member because 25 = 5*5 and 25+5+5 = 5*7 is also a semiprime.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A001358.

Contains A108570.

Programs

Maple

N:= 1000:
Primes:= select(isprime, [2,seq(i,i=3..N)]):
SP:= sort([seq(seq([p,q],q=select(t -> t >= p and p*t<=N, Primes)),p=Primes)],(a,b) -> a[1]*a[2] t[1]*t[2], select(t -> numtheory:-bigomega(t[1]*t[2]+t[1]+t[2])=2, SP));

Mathematica

Select[Union@ Apply[Join, Table[Flatten@{p #, Sort[{p, #}]} & /@ Prime@ Range@ PrimePi@ Floor[Max[#]/p], {p, #}]] &@ Prime@ Range@ 97, PrimeOmega[Total@ #] == 2 &][[All, 1]] (* Michael De Vlieger, Dec 15 2019 *)

PARI

issemi(n)=bigomega(n)==2
list(lim)=my(v=List()); forprime(p=2, sqrtint(lim\=1), forprime(q=p, lim\p, if(issemi(p*q+p+q), listput(v,p*q)))); Set(v) \\ Charles R Greathouse IV, Dec 16 2019

Python

from sympy import factorint
def is_semiprime(n): return sum(e for e in factorint(n).values()) == 2
def ok(n):
    f = factorint(n, multiple=True)
    if len(f) != 2: return False
    p, q = f
    return len(factorint(p*q + p + q, multiple=True)) == 2
print(list(filter(ok, range(506)))) # Michael S. Branicky, Sep 22 2021