A330203 Composite numbers k such that D(k) == 3 (mod k), where D(k) is the k-th central Delannoy number (A001850).

10, 15, 50, 370, 2418, 4371, 5341, 8430, 20535, 25338, 26958, 278674, 1194649, 4304445, 11984885, 12327121, 20746461, 27585010, 72363853, 79501818

Offset: 1

Author

Amiram Eldar, Dec 05 2019

Keywords

• nonn
• more

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.

Examples

			10 is in the sequence since it is composite and D(10) = 8097453 == 3 (mod 10).
		

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 Allouche and Guentcho Skordev, 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.

Crossrefs

Cf. A001850, A008316.

Programs

Mathematica

Select[Range[2500], CompositeQ[#] && Divisible[LegendreP[#, 3] - 3, #] &]

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

Extensions

a(18) from Robin Visser, Aug 08 2023

a(19)-a(20) from Robin Visser, Sep 11 2023