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
Keywords
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.
Programs
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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] -
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);}
Comments