A353056 Composite numbers of the form k^2+k+1 all of whose prime factors are of that same form.

21, 91, 273, 343, 507, 651, 1333, 4557, 6321, 6643, 27391, 36673, 50851, 65793, 83811, 105301, 139503, 190533, 194923, 217623, 234741, 391251, 545383, 1647373, 1961401, 2032051, 2376223, 4517751, 6118203, 6484663, 11590621, 13180531, 14535157, 20155611, 28371603, 35646871

Offset: 1

Author

Michel Marcus, Apr 20 2022

Keywords

• nonn

Examples

			21 = 4^2+4+1 and its factors are 3 and 7, terms of A002383. So 21 is a term.
		

Links

• David A. Corneth, Table of n, a(n) for n = 1..318
• Cody S. Hansen and Pace P. Nielsen, Prime factors of phi3(x) of the same form, arXiv:2204.08971 [math.NT], 2022.

Crossrefs

Subsequence of A174969.

Cf. A002383.

Programs

Maple

q:= n-> not isprime(n) and andmap(p-> issqr(4*p-3), numtheory[factorset](n)):
select(q, [k*(k+1)+1$k=4..6000])[];  # Alois P. Heinz, Apr 20 2022

Mathematica

Select[Table[n^2 + n + 1, {n, 1, 6000}], CompositeQ[#] && AllTrue[FactorInteger[#][[;; , 1]], IntegerQ@Sqrt[4*#1 - 3] &] &] (* Amiram Eldar, Apr 20 2022 *)

PARI

lista(nn) = {for (n=1, nn, my(x=n^2+n+1); if (! isprime(x), my(fa=factor(x), ok=1); for (k=1, #fa~, my(fk=fa[k,1]); if (! issquare(4*fk-3), ok = 0);); if (ok, print1(x, ", "));););}

Python

from sympy import isprime, factorint
from itertools import count, takewhile
def agento(N): # generator of terms up to limit N
    form = set(takewhile(lambda x: x<=N, (k**2 + k + 1 for k in count(1))))
    for t in sorted(form):
        if not isprime(t) and all(p in form for p in factorint(t)):
            yield t
print(list(agento(10**8))) # Michael S. Branicky, Apr 20 2022