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

1,1

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

Table of n, a(n) for n=1..50.

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

Cf. A002426.

Sequence in context: A187408 A211055 A336025 * A136769 A351600 A115075

Adjacent sequences: A308479 A308480 A308481 * A308483 A308484 A308485

Keyword

nonn

Author

Felix Fröhlich, May 30 2019

Extensions

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

Status

approved