OFFSET
1,1
COMMENTS
Equivalently, composite numbers k such that P(k, 3) == 3 (mod k), where P(k, 3) = D(k) is the k-th Legendre polynomial evaluated at 3.
P(p, 3) == 3 (mod p) for all primes p. This is a special case of Schur congruences, named after Issai Schur, first published by his student Hildegard Ille in her Ph.D. thesis in 1924, and proven by Wahab in 1952. This sequence consists of the composite numbers for which the congruence holds.
REFERENCES
Hildegard Ille, Zur Irreduzibilität der Kugelfunktionen, Jahrbuch der Dissertationen der Universität Berlin, (1924).
Peter S. Landweber, Elliptic Curves and Modular Forms in Algebraic Topology: Proceedings of a Conference held at the Institute for Advanced Study, Princeton, Sept. 15-17, 1986, Springer, 2006. See pp. 74-76.
LINKS
Jean-Paul Allouchea and Guentcho Skordevb, Schur congruences, Carlitz sequences of polynomials and automaticity, Discrete Mathematics, Vol 214 (2000), pp. 21-49.
S. K. Chatterjea, On Congruence Properties of Legendre Polynomials, Mathematics Magazine, Vol. 34, No. 6 (1961), pp. 329-336.
Sen-Peng Eu, Shu-Chung Liu, and Yeong-Nan Yeh, On the Congruences of Some Combinatorial Numbers, Studies in Applied Mathematics, Vol. 116, No. 2 (2006), pp. 135-144.
J. H. Wahab, New cases of irreducibility for Legendre polynomials, Duke Mathematical Journal, Vol. 19 (1952), pp. 165-176.
EXAMPLE
10 is in the sequence since it is composite and D(10) = 8097453 == 3 (mod 10).
MATHEMATICA
Select[Range[2500], CompositeQ[#] && Divisible[LegendreP[#, 3] - 3, #] &]
PROG
(Sage)
a, b = 1, 1
for n in range(1, 10000):
a, b = b, ((6*n-3)*b - (n-1)*a)//n
if (b%n == 3) and (not Integer(n).is_prime()): print(n) # Robin Visser, Aug 08 2023
CROSSREFS
KEYWORD
nonn,more
AUTHOR
Amiram Eldar, Dec 05 2019
EXTENSIONS
a(18) from Robin Visser, Aug 08 2023
a(19)-a(20) from Robin Visser, Sep 11 2023
STATUS
approved