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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. 0
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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified September 26 05:46 EDT 2022. Contains 356986 sequences. (Running on oeis4.)