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.

6, 15, 21, 22, 105, 119, 231, 426, 483, 1290, 1939, 4429, 4450, 4578, 10609, 12999, 14118, 16899, 23262, 26733, 37401, 39858, 82194, 108345, 121335, 127434, 302253, 380757, 724647, 836437, 840147, 1078270, 1522677, 2007411, 15009050, 28913991

Offset: 1

Author

Amiram Eldar, Dec 05 2019

Keywords

• nonn
• more

Comments

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.

Examples

			6 is in the sequence since it is composite and P(6, 7) = 1651609 == 7 (mod 6).
		

Crossrefs

Cf. A008316, A084768, A330203.

Programs

Mathematica

Select[Range[2000], CompositeQ[#] && Divisible[LegendreP[#, 7] - 7, #] &]

PARI

isok(k) = Mod(subst(pollegendre(k), x, 7), k) == 7;
forcomposite (k=1, 10000, if (isok(k), print1(k, ", "))); \\ Michel Marcus, Dec 06 2019

Sage

a, b = 1, 7
for n in range(2, 10000):
    a, b = b, ((14*n-7)*b - (n-1)*a)//n
    if (b%n == 7%n) and (not Integer(n).is_prime()): print(n)  # Robin Visser, Aug 18 2023

Extensions

a(35)-a(36) from Robin Visser, Aug 18 2023