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A353056
Composite numbers of the form k^2+k+1 all of whose prime factors are of that same form.
1
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
1,1
LINKS
Cody S. Hansen and Pace P. Nielsen, Prime factors of phi3(x) of the same form, arXiv:2204.08971 [math.NT], 2022.
EXAMPLE
21 = 4^2+4+1 and its factors are 3 and 7, terms of A002383. So 21 is a term.
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 *)
PROG
(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
CROSSREFS
Subsequence of A174969.
Cf. A002383.
Sequence in context: A020248 A225705 A259758 * A203173 A194532 A065827
KEYWORD
nonn
AUTHOR
Michel Marcus, Apr 20 2022
STATUS
approved