A308482 Composites c such that T_{c-1} == (c/3)*3^(c-1) (mod c), where T_i denotes the i-th central trinomial coefficient (A002426) and (/) denotes the Kronecker symbol.

4, 9, 20, 25, 27, 40, 49, 80, 81, 121, 169, 189, 243, 272, 289, 361, 369, 400, 416, 470, 529, 544, 567, 729, 841, 961, 1071, 1323, 1369, 1539, 1681, 1849, 2000, 2187, 2209, 2809, 2889, 3213, 3481, 3721, 4489, 4617, 5041, 5329, 6241, 6561, 6889, 7749, 7921, 8667

Offset: 1

Author

Felix Fröhlich, May 30 2019

Keywords

• nonn

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.

Crossrefs

Cf. A002426.

Programs

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 *)

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, ", ")))

Extensions

a(45)-a(50) from Amiram Eldar, Jul 10 2019