A347008 - OEIS

%I #13 Aug 10 2021 22:01:16

%S 23,47,119,167,179,323,407,419,527,587,639,647,879,935,1043,1103,1119,

%T 1139,1215,1223,1247,1271,1331,1367,1403,1455,1595,1599,1631,1691,

%U 1775,1791,1859,1895,1931,1943,1959,1967,1979,2099,2111,2175,2183,2219,2231,2435,2471,2483,2495,2543,2559,2603

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

%H Robert Israel, <a href="/A347008/b347008.txt">Table of n, a(n) for n = 1..5000</a>

%e 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.

%p N:= 10000: # to produce terms <= N

%p R:= Vector(N):

%p P:= select(isprime, [2,seq(i,i=3..N/3,2)]):

%p for i from 1 to nops(P) do

%p   for j from 1 to i-1 do

%p    v:=P[i]*P[j]+P[i]+P[j];

%p    if v <= N then R[v]:= R[v]+1 fi

%p od od:

%p select(t -> R[t]=2, [$1..N]);

%o (Python)

%o from sympy import primerange

%o from collections import Counter

%o def aupto(limit):

%o     primes = list(primerange(2, limit//3+1))

%o     nums = [p*q+p+q for i, p in enumerate(primes) for q in primes[i+1:]]

%o     counts = Counter([k for k in nums if k <= limit])

%o     return sorted(k for k in counts if counts[k] == 2)

%o print(aupto(2604)) # _Michael S. Branicky_, Aug 10 2021

%Y Cf. A198277.

%K nonn

%O 1,1

%A _J. M. Bergot_ and _Robert Israel_, Aug 10 2021