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A373763
Numbers k that are neither primes nor squares of primes such that A006134(k) - A102283(k) is divisible by k.
1
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
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.
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
Sequence in context: A107580 A138610 A128530 * A044129 A044510 A338556
KEYWORD
nonn
AUTHOR
Amiram Eldar, Jun 18 2024
STATUS
approved