OFFSET
1,1
COMMENTS
Composites satisfying a weaker version of an analog to a congruence satisfied by all primes > 3 (cf. Cao, Sun, 2015, Theorem 1.1 (i); cf. Sun, 2011, Remark to Conjecture A69).
Up to 9000, 189 is the only composite satisfying the congruence modulo c^2. Do any other such composites exist?
LINKS
Hui-Qin Cao and Zhi-Wei Sun, Some congruences involving binomial coefficients, Colloquium Mathematicum 139 (2015), 127-136, arXiv:1006.3069 [math.NT], 2010-2015.
Zhi-Wei Sun, Open Conjectures on Congruences, arXiv:0911.5665 [math.NT], 2011.
MATHEMATICA
aQ[n_] := CompositeQ[n] && Divisible[3^(n-1)*(Hypergeometric2F1[1/2, 1-n, 1, 4/3] - JacobiSymbol[n, 3]) , n]; Select[Range[1000], aQ] (* Amiram Eldar, Jul 10 2019 *)
PROG
(PARI) t(n) = sum(k=0, floor(n/2), binomial(n, k)*binomial(n-k, k))
is(n) = Mod(t(n-1), n)==kronecker(n, 3)*3^(n-1)
forcomposite(c=1, , if(is(c), print1(c, ", ")))
CROSSREFS
KEYWORD
nonn
AUTHOR
Felix Fröhlich, May 30 2019
EXTENSIONS
a(45)-a(50) from Amiram Eldar, Jul 10 2019
STATUS
approved