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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.
0
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.
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
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