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%I #37 Dec 20 2018 02:01:48
%S 283686649,4514260853041
%N Composite numbers n such that binomial(2n-1, n-1) == 1 (mod n^2).
%C Babbage proved the congruence holds if n > 2 is prime.
%C See A088164 and A263882 for references, links, and additional comments.
%C Conjecture: n is a term if and only if n = A088164(i)^2 for some i >= 1 (cf. McIntosh, 1995, p. 385). - _Felix Fröhlich_, Jan 27 2016
%C The "if" part of the conjecture is true: see the McIntosh reference. - _Jonathan Sondow_, Jan 28 2016
%C The above conjecture implies that this sequence and A228562 are disjoint. - _Felix Fröhlich_, Jan 27 2016
%C Composites c such that A281302(c) > 1. - _Felix Fröhlich_, Feb 21 2018
%H Richard J. McIntosh, <a href="http://matwbn.icm.edu.pl/ksiazki/aa/aa71/aa7144.pdf">On the converse of Wolstenholme's Theorem</a>, Acta Arithmetica, 71 (1995), 381-389.
%H J. Sondow, Extending Babbage's (non-)primality tests, in <a href="https://doi.org/10.1007/978-3-319-68032-3_19">Combinatorial and Additive Number Theory II</a>, Springer Proc. in Math. & Stat., Vol. 220, 269-277, CANT 2015 and 2016, New York, 2017; <a href="http://arxiv.org/abs/1812.07650">arXiv:1812.07650 [math.NT]</a>, 2018.
%e a(1) = 16843^2 and a(2) = 2124679^2 are squares of Wolstenholme primes A088164.
%Y Cf. A000984, A034602, A082180, A088164, A099905, A099906, A099907, A099908, A136327, A177783, A212557, A228562, A242473, A244214, A244919, A246130, A246132, A246133, A246134, A260209, A260210, A263429, A263882, A281302.
%K nonn,bref,hard,more
%O 1,1
%A _Jonathan Sondow_, Jan 25 2016
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