A373763 Numbers k that are neither primes nor squares of primes such that A006134(k) - A102283(k) is divisible by k.

1, 27, 63, 81, 238, 243, 247, 279, 322, 580, 671, 729, 1222, 2074, 2187, 3172, 3550, 3577, 4185, 5589, 6561, 7805, 7957, 8239, 8701, 8890, 9040, 9064, 9523, 9730, 9898, 10087, 10138, 10549, 11074, 11176, 11440, 11473, 11920, 12232, 12430, 12604, 13900, 14287, 14410

Offset

1,2

Comments

The congruence A006134(k) == A102283(k) (mod k) holds for all values of k that are primes or squares of primes.

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

Cf. A006134, A102283.

Sequence in context: A107580 A138610 A128530 * A044129 A044510 A338556

Adjacent sequences: A373760 A373761 A373762 * A373764 A373765 A373766

Keyword

nonn

Author

Amiram Eldar, Jun 18 2024

Status

approved