A347008 Numbers that can be written in exactly two ways as p*q+p+q where p and q are primes with p < q.

23, 47, 119, 167, 179, 323, 407, 419, 527, 587, 639, 647, 879, 935, 1043, 1103, 1119, 1139, 1215, 1223, 1247, 1271, 1331, 1367, 1403, 1455, 1595, 1599, 1631, 1691, 1775, 1791, 1859, 1895, 1931, 1943, 1959, 1967, 1979, 2099, 2111, 2175, 2183, 2219, 2231, 2435, 2471, 2483, 2495, 2543, 2559, 2603

Offset: 1

Author

J. M. Bergot and Robert Israel, Aug 10 2021

Keywords

• nonn

Examples

			a(3) = 119 is a term because 119 = 5*19+5+19 = 3*29+3+29 are the two ways to produce 119 = p*q+p+q with primes p < q.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..5000

Crossrefs

Cf. A198277.

Programs

Maple

N:= 10000: # to produce terms <= N
R:= Vector(N):
P:= select(isprime, [2,seq(i,i=3..N/3,2)]):
for i from 1 to nops(P) do
  for j from 1 to i-1 do
   v:=P[i]*P[j]+P[i]+P[j];
   if v <= N then R[v]:= R[v]+1 fi
od od:
select(t -> R[t]=2, [$1..N]);

Python

from sympy import primerange
from collections import Counter
def aupto(limit):
    primes = list(primerange(2, limit//3+1))
    nums = [p*q+p+q for i, p in enumerate(primes) for q in primes[i+1:]]
    counts = Counter([k for k in nums if k <= limit])
    return sorted(k for k in counts if counts[k] == 2)
print(aupto(2604)) # Michael S. Branicky, Aug 10 2021