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

%I A330203 #33 Jun 29 2025 15:29:04
%S A330203 10,15,50,370,2418,4371,5341,8430,20535,25338,26958,278674,1194649,
%T A330203 4304445,11984885,12327121,20746461,27585010,72363853,79501818
%N A330203 Composite numbers k such that D(k) == 3 (mod k), where D(k) is the k-th central Delannoy number (A001850).
%C A330203 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.
%C A330203 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.
%D A330203 Hildegard Ille, Zur Irreduzibilität der Kugelfunktionen, Jahrbuch der Dissertationen der Universität Berlin, (1924).
%D A330203 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.
%H A330203 Jean-Paul Allouche and Guentcho Skordev, <a href="https://doi.org/10.1016/S0012-365X(99)00195-8">Schur congruences, Carlitz sequences of polynomials and automaticity</a>, Discrete Mathematics, Vol 214 (2000), pp. 21-49.
%H A330203 S. K. Chatterjea, <a href="http://doi.org/10.2307/2688354">On Congruence Properties of Legendre Polynomials</a>, Mathematics Magazine, Vol. 34, No. 6 (1961), pp. 329-336.
%H A330203 Sen-Peng Eu, Shu-Chung Liu, and Yeong-Nan Yeh, <a href="https://doi.org/10.1111/j.1467-9590.2006.00337.x">On the Congruences of Some Combinatorial Numbers</a>, Studies in Applied Mathematics, Vol. 116, No. 2 (2006), pp. 135-144.
%H A330203 J. H. Wahab, <a href="https://doi.org/10.1215/S0012-7094-52-01917-0">New cases of irreducibility for Legendre polynomials</a>, Duke Mathematical Journal, Vol. 19 (1952), pp. 165-176.
%e A330203 10 is in the sequence since it is composite and D(10) = 8097453 == 3 (mod 10).
%t A330203 Select[Range[2500], CompositeQ[#] && Divisible[LegendreP[#, 3] - 3, #] &]
%o A330203 (Sage)
%o A330203 a, b = 1, 1
%o A330203 for n in range(1, 10000):
%o A330203     a, b = b, ((6*n-3)*b - (n-1)*a)//n
%o A330203     if (b%n == 3) and (not Integer(n).is_prime()): print(n)  # _Robin Visser_, Aug 08 2023
%Y A330203 Cf. A001850, A008316.
%K A330203 nonn,more
%O A330203 1,1
%A A330203 _Amiram Eldar_, Dec 05 2019
%E A330203 a(18) from _Robin Visser_, Aug 08 2023
%E A330203 a(19)-a(20) from _Robin Visser_, Sep 11 2023