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A330204 Composite numbers k such that P(k, 5) == 5 (mod k), where P(k, 5) = A006442(k) is the k-th Legendre polynomial evaluated at 5.

%I A330204 #22 Sep 11 2023 18:37:07
%S A330204 4,15,35,165,255,615,1815,1876,2636,2948,5380,5565,11235,28545,288380,
%T A330204 903644,1807995,2486165,2674060,10538572,11791595,14145121,28558415,
%U A330204 45153277,45682751
%N A330204 Composite numbers k such that P(k, 5) == 5 (mod k), where P(k, 5) = A006442(k) is the k-th Legendre polynomial evaluated at 5.
%C A330204 P(p, 5) == 5 (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 A330204 4 is in the sequence since it is composite and P(4, 5) = 2641 == 5 (mod 4).
%t A330204 Select[Range[3000], CompositeQ[#] && Divisible[LegendreP[#, 5] - 5, #] &]
%o A330204 (PARI) isok(k) = Mod(subst(pollegendre(k), x, 5), k) == 5;
%o A330204 forcomposite (k=1, 10000, if (isok(k), print1(k, ", "))); \\ _Michel Marcus_, Dec 06 2019
%o A330204 (Sage)
%o A330204 a, b = 1, 5
%o A330204 for n in range(2, 10000):
%o A330204     a, b = b, ((10*n-5)*b - (n-1)*a)//n
%o A330204     if (b%n == 5%n) and (not Integer(n).is_prime()): print(n)  # _Robin Visser_, Aug 17 2023
%Y A330204 Cf. A006442, A008316, A330203.
%K A330204 nonn,more
%O A330204 1,1
%A A330204 _Amiram Eldar_, Dec 05 2019
%E A330204 a(22)-a(23) from _Robin Visser_, Aug 17 2023
%E A330204 a(24)-a(25) from _Robin Visser_, Sep 11 2023