

A330203


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


2



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

1,1


COMMENTS

Equivalently, composite numbers k such that P(k, 3) == 3 (mod k), where P(k, 3) = D(k) is the kth 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. 1517, 1986, Springer, 2006. See pp. 7476.


LINKS

Table of n, a(n) for n=1..17.
JeanPaul Allouchea and Guentcho Skordevb, Schur congruences, Carlitz sequences of polynomials and automaticity, Discrete Mathematics, Vol 214 (2000), pp. 2149.
S. K. Chatterjea, On Congruence Properties of Legendre Polynomials, Mathematics Magazine, Vol. 34, No. 6 (1961), pp. 329336.
SenPeng Eu, ShuChung Liu, and YeongNan Yeh, On the Congruences of Some Combinatorial Numbers, Studies in Applied Mathematics, Vol. 116, No. 2 (2006), pp. 135144.
J. H. Wahab, New cases of irreducibility for Legendre polynomials, Duke Mathematical Journal, Vol. 19 (1952), pp. 165176.


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, #] &]


CROSSREFS

Cf. A001850, A008316.
Sequence in context: A212794 A048061 A188651 * A272307 A092192 A119039
Adjacent sequences: A330200 A330201 A330202 * A330204 A330205 A330206


KEYWORD

nonn,more


AUTHOR

Amiram Eldar, Dec 05 2019


STATUS

approved



