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A330203
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Composite numbers k such that D(k) == 3 (mod k), where D(k) is the k-th central Delannoy number (A001850).
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2
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10, 15, 50, 370, 2418, 4371, 5341, 8430, 20535, 25338, 26958, 278674, 1194649, 4304445, 11984885, 12327121, 20746461
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OFFSET
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1,1
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COMMENTS
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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.
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REFERENCES
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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.
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LINKS
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Table of n, a(n) for n=1..17.
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.
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EXAMPLE
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10 is in the sequence since it is composite and D(10) = 8097453 == 3 (mod 10).
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MATHEMATICA
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Select[Range[2500], CompositeQ[#] && Divisible[LegendreP[#, 3] - 3, #] &]
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CROSSREFS
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Cf. A001850, A008316.
Sequence in context: A212794 A048061 A188651 * A272307 A092192 A119039
Adjacent sequences: A330200 A330201 A330202 * A330204 A330205 A330206
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KEYWORD
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nonn,more
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AUTHOR
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Amiram Eldar, Dec 05 2019
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STATUS
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approved
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