A330203 -id:A330203 - cp's OEIS frontend

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

Amiram Eldar, Dec 05 2019

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.

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

Cf. A006442, A008316, A330203.

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

isok(k) = Mod(subst(pollegendre(k), x, 5), k) == 5;
forcomposite (k=1, 10000, if (isok(k), print1(k, ", "))); \\ Michel Marcus, Dec 06 2019

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

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

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

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.

Original entry on oeis.org

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

Amiram Eldar, Dec 05 2019

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.

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

Cf. A008316, A084768, A330203.

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

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

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

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

cp's OEIS Frontend

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.

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