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.

4, 15, 35, 165, 255, 615, 1815, 1876, 2636, 2948, 5380, 5565, 11235, 28545, 288380, 903644, 1807995, 2486165, 2674060, 10538572, 11791595, 14145121, 28558415, 45153277, 45682751

Offset

1,1

Comments

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.

Links

Table of n, a(n) for n=1..25.

Example

4 is in the sequence since it is composite and P(4, 5) = 2641 == 5 (mod 4).

Mathematica

Select[Range[3000], CompositeQ[#] && Divisible[LegendreP[#, 5] - 5, #] &]

Prog

(PARI) isok(k) = Mod(subst(pollegendre(k), x, 5), k) == 5;
forcomposite (k=1, 10000, if (isok(k), print1(k, ", "))); \\ Michel Marcus, Dec 06 2019
(Sage)
a, b = 1, 5
for n in range(2, 10000):
    a, b = b, ((10*n-5)*b - (n-1)*a)//n
    if (b%n == 5%n) and (not Integer(n).is_prime()): print(n)  # Robin Visser, Aug 17 2023

Crossrefs

Cf. A006442, A008316, A330203.

Sequence in context: A213420 A209345 A323452 * A190093 A077414 A304487

Adjacent sequences: A330201 A330202 A330203 * A330205 A330206 A330207

Keyword

nonn,more

Author

Amiram Eldar, Dec 05 2019

Extensions

a(22)-a(23) from Robin Visser, Aug 17 2023

a(24)-a(25) from Robin Visser, Sep 11 2023

Status

approved