A267824 Composite numbers n such that binomial(2n-1, n-1) == 1 (mod n^2).

283686649, 4514260853041

Offset: 1

Author

Jonathan Sondow, Jan 25 2016

Keywords

• nonn
• bref
• hard
• more

Comments

Babbage proved the congruence holds if n > 2 is prime.

See A088164 and A263882 for references, links, and additional comments.

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

The "if" part of the conjecture is true: see the McIntosh reference. - Jonathan Sondow, Jan 28 2016

The above conjecture implies that this sequence and A228562 are disjoint. - Felix Fröhlich, Jan 27 2016

Composites c such that A281302(c) > 1. - Felix Fröhlich, Feb 21 2018

Examples

			a(1) = 16843^2 and a(2) = 2124679^2 are squares of Wolstenholme primes A088164.
		

Links

• Richard J. McIntosh, On the converse of Wolstenholme's Theorem, Acta Arithmetica, 71 (1995), 381-389.
• J. Sondow, Extending Babbage's (non-)primality tests, in Combinatorial and Additive Number Theory II, Springer Proc. in Math. & Stat., Vol. 220, 269-277, CANT 2015 and 2016, New York, 2017; arXiv:1812.07650 [math.NT], 2018.

Crossrefs

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.