OFFSET
1,2
COMMENTS
LINKS
Amiram Eldar, Table of n, a(n) for n = 1..1000
Moa Apagodu and Doron Zeilberger, Using the "Freshman's Dream" to Prove Combinatorial Congruences, The American Mathematical Monthly, Vol. 124, No. 7 (2017), pp. 597-608; arXiv preprint, arXiv:1606.03351 [math.CO], 2016.
Ji-Cai Liu, Supercongruences involving Motzkin numbers and central trinomial coefficients, arXiv:2208.10275 [math.NT], 2022.
Zhi-Wei Sun and Roberto Tauraso, On some new congruences for binomial coefficients, International Journal of Number Theory, Vol. 7, No. 3 (2011), pp. 645-662; arXiv preprint, arXiv:0709.1665 [math.NT], 2007-2011.
MATHEMATICA
q[n_] := !PrimeQ[n] && !PrimeQ[Sqrt[n]] && Divisible[Sum[Binomial[2*k, k], {k, 0, n - 1}] - JacobiSymbol[n, 3], n]; Select[Range[1000], q]
PROG
(PARI) is1(k) = !isprime(k) && !(issquare(k) && isprime(sqrtint(k)));
lista(kmax) = {my(s0 = 1, s1 = 3); print1(1, ", "); for(k = 2, kmax, s2 = ((5*k - 2) * s1 - 2 * (2*k - 1) * s0 )/k; if(is1(k + 1) && !((s2 - [1, -1, 0][k % 3 + 1]) % (k + 1)), print1(k + 1, ", ")); s0 = s1; s1 = s2); }
CROSSREFS
KEYWORD
nonn
AUTHOR
Amiram Eldar, Jun 18 2024
STATUS
approved