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A351579 Primes p such that the sum of p and the next two primes is the product of two consecutive primes.

%I A351579 #16 Feb 20 2022 06:44:37
%S A351579 3,43,3671,51473,53051,64811,71143,121591,137383,154111,161459,228521,
%T A351579 284573,344053,433141,544403,679709,702743,767071,995303,1158139,
%U A351579 1267481,1301507,1320023,1342667,1512293,1682987,1839221,1982891,2022101,2174287,2198153,2370943,2403061,2770549,4148923,4368121
%N A351579 Primes p such that the sum of p and the next two primes is the product of two consecutive primes.
%C A351579 A000040(k) for k such that A034961(k) is in A006094.
%C A351579 The Generalized Bunyakovsky Conjecture implies that, for example, there are infinitely many terms of the form 12*s^2+12*s-1 where the next two primes are 12*s^2+12*s+1 and 12*s^2+12*s+5 and the sum of these is (6*s+1)*(6*s+5).
%H A351579 Robert Israel, <a href="/A351579/b351579.txt">Table of n, a(n) for n = 1..120</a>
%e A351579 a(3) = 3671 is a term because 3671, 3673, 3677 are three consecutive primes with 3671+3673+3677 = 11021 = 103*107 and 103 and 107 are two consecutive primes.
%p A351579 q:= proc(n) local r,s;
%p A351579   r:= nextprime(floor(sqrt(n)));
%p A351579   s:= n/r;
%p A351579   s::integer and s = prevprime(r)
%p A351579 end proc:
%p A351579 P:= select(isprime,[2,seq(i,i=3..10^7)]):
%p A351579 S:= [0,op(ListTools:-PartialSums(P))]:
%p A351579 map(t -> P[t], select(i -> q(S[i+3]-S[i]), [$1..nops(S)-3]));
%t A351579 prodQ[n_] := Module[{f = FactorInteger[n]}, f[[;; , 2]] == {1, 1} && f[[2, 1]] == NextPrime[f[[1, 1]]]]; q[p_] := PrimeQ[p] && prodQ[p + Plus @@ NextPrime[p, {1, 2}]]; Select[Range[5*10^6], q] (* _Amiram Eldar_, Feb 14 2022 *)
%o A351579 (PARI) isok(p) = {if (isprime(p), my(q=nextprime(p+1), f=factor(p+q+nextprime(q+1))); (omega(f) == 2) && (bigomega(f) == 2) && (f[2,1] == nextprime(f[1,1]+1)););} \\ _Michel Marcus_, Feb 14 2022
%Y A351579 Cf. A000040, A006094, A034961.
%K A351579 nonn
%O A351579 1,1
%A A351579 _J. M. Bergot_ and _Robert Israel_, Feb 13 2022