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A330205 Composite numbers k such that P(k, 7) == 7 (mod k), where P(k, 7) = A084768(k) is the k-th Legendre polynomial evaluated at 7.

%I A330205 #15 Aug 18 2023 15:05:18
%S A330205 6,15,21,22,105,119,231,426,483,1290,1939,4429,4450,4578,10609,12999,
%T A330205 14118,16899,23262,26733,37401,39858,82194,108345,121335,127434,
%U A330205 302253,380757,724647,836437,840147,1078270,1522677,2007411,15009050,28913991
%N A330205 Composite numbers k such that P(k, 7) == 7 (mod k), where P(k, 7) = A084768(k) is the k-th Legendre polynomial evaluated at 7.
%C A330205 P(p, 7) == 7 (mod p) for all primes p. This is a special case of Schur congruences (see A330203 for references). This sequence consists of the composite numbers for which the congruence holds.
%e A330205 6 is in the sequence since it is composite and P(6, 7) = 1651609 == 7 (mod 6).
%t A330205 Select[Range[2000], CompositeQ[#] && Divisible[LegendreP[#, 7] - 7, #] &]
%o A330205 (PARI) isok(k) = Mod(subst(pollegendre(k), x, 7), k) == 7;
%o A330205 forcomposite (k=1, 10000, if (isok(k), print1(k, ", "))); \\ _Michel Marcus_, Dec 06 2019
%o A330205 (Sage)
%o A330205 a, b = 1, 7
%o A330205 for n in range(2, 10000):
%o A330205     a, b = b, ((14*n-7)*b - (n-1)*a)//n
%o A330205     if (b%n == 7%n) and (not Integer(n).is_prime()): print(n)  # _Robin Visser_, Aug 18 2023
%Y A330205 Cf. A008316, A084768, A330203.
%K A330205 nonn,more
%O A330205 1,1
%A A330205 _Amiram Eldar_, Dec 05 2019
%E A330205 a(35)-a(36) from _Robin Visser_, Aug 18 2023