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A066938 Primes of the form p*q+p+q, where p and q are primes.

%I A066938 #35 Jan 02 2022 11:25:22
%S A066938 11,17,23,31,41,47,53,59,71,79,83,89,107,113,127,131,151,167,179,191,
%T A066938 227,239,251,263,269,271,293,311,359,383,419,431,439,443,449,479,491,
%U A066938 503,521,587,593,599,607,631,647,659,683,701,719,727,743,773,809,827
%N A066938 Primes of the form p*q+p+q, where p and q are primes.
%C A066938 For p not equal to q, either p*q or p+q is odd, so their sum is odd.
%C A066938 The representation is ambiguous, e.g. 2*7+2+7 = 23 = 3*5+3+5.
%C A066938 Complement of A198273 with respect to A000040. - _Reinhard Zumkeller_, Oct 23 2011
%C A066938 None of these primes are in A158913 since if p*q+p+q is a prime, then sigma(p*q+p+q) = sigma(p*q). - _Amiram Eldar_, Nov 15 2021
%H A066938 Reinhard Zumkeller, <a href="/A066938/b066938.txt">Table of n, a(n) for n = 1..10000</a>
%F A066938 A067432(A049084(a(n))) > 0. - _Reinhard Zumkeller_, Oct 23 2011
%F A066938 A054973(a(n)+1) >= 2. - _Amiram Eldar_, Nov 15 2021
%e A066938 59 is in the sequence because 59 = 2 * 19 + 2 + 19.
%t A066938 nn = 1000; n2 = PrimePi[nn/3]; Select[Union[Flatten[Table[(Prime[i] + 1) (Prime[j] + 1) - 1, {i, n2}, {j, n2}]]], # <= nn && PrimeQ[#] &]
%o A066938 (Haskell)
%o A066938 a066938 n = a066938_list !! (n-1)
%o A066938 a066938_list = map a000040 $ filter ((> 0) . a067432) [1..]
%o A066938 -- _Reinhard Zumkeller_, Oct 23 2011
%o A066938 (PARI) is(n)=fordiv(n+1,d,my(p=d-1,q=(n+1)/d-1); if(isprime(p) && isprime(q), return(isprime(n)))); 0 \\ _Charles R Greathouse IV_, Jul 23 2013
%Y A066938 Cf. A054973, A067432, A072668, A072673, A088709, A158913, A198273, A198277.
%K A066938 nonn
%O A066938 1,1
%A A066938 _Reinhard Zumkeller_, Jan 24 2002
%E A066938 Edited by _Robert G. Wilson v_, Feb 01 2002