A351579 - OEIS

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

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OFFSET

1,1

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).

LINKS

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

EXAMPLE

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.

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 *)

PROG

(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

CROSSREFS

Cf. A000040, A006094, A034961.

Sequence in context: A009720 A300873 A333329 * A201173 A290777 A309401

Adjacent sequences: A351576 A351577 A351578 * A351580 A351581 A351582

KEYWORD

nonn

AUTHOR

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

STATUS

approved