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

283686649, 4514260853041

Offset

1,1

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

Links

Table of n, a(n) for n=1..2.

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.

Example

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

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.

Sequence in context: A017362 A017482 A017614 * A185428 A320878 A320879

Adjacent sequences: A267821 A267822 A267823 * A267825 A267826 A267827

Keyword

nonn,bref,hard,more

Author

Jonathan Sondow, Jan 25 2016

Status

approved