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

3, 43, 3671, 51473, 53051, 64811, 71143, 121591, 137383, 154111, 161459, 228521, 284573, 344053, 433141, 544403, 679709, 702743, 767071, 995303, 1158139, 1267481, 1301507, 1320023, 1342667, 1512293, 1682987, 1839221, 1982891, 2022101, 2174287, 2198153, 2370943, 2403061, 2770549, 4148923, 4368121

Offset: 1

Author

J. M. Bergot and Robert Israel, Feb 13 2022

Keywords

• nonn

Comments

A000040(k) for k such that A034961(k) is in A006094.

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).

Examples

			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.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..120

Crossrefs

Cf. A000040, A006094, A034961.

Programs

Maple

q:= proc(n) local r,s;
  r:= nextprime(floor(sqrt(n)));
  s:= n/r;
  s::integer and s = prevprime(r)
end proc:
P:= select(isprime,[2,seq(i,i=3..10^7)]):
S:= [0,op(ListTools:-PartialSums(P))]:
map(t -> P[t], select(i -> q(S[i+3]-S[i]), [$1..nops(S)-3]));

Mathematica

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 *)

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