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

1,1

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.

Links

Table of n, a(n) for n=1..36.

Example

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

Mathematica

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

Prog

(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

Crossrefs

Cf. A008316, A084768, A330203.

Sequence in context: A208542 A012412 A009092 * A015793 A261078 A063466

Adjacent sequences: A330202 A330203 A330204 * A330206 A330207 A330208

Keyword

nonn,more

Author

Amiram Eldar, Dec 05 2019

Extensions

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

Status

approved