A373764 Odd composite numbers k such that A086615(k-1) == 2*(-1)^((k-1)/2) (mod k).

369, 441, 4473, 10609, 292559

Offset: 1

Author

Amiram Eldar, Jun 18 2024

Keywords

• nonn
• hard
• more

Comments

The congruence holds for all odd primes.

a(6) > 2.3*10^7, if it exists.

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.

Crossrefs

Cf. A086615.

Programs

Mathematica

seq[kmax_] := Module[{s0 = 1, s1 = 2, s2 = 4, s3, s = {}}, Do[s3 = (3*(k+1)*s2 + (k-4)*s1 - 3*(k-1)*s0)/(k+2); If[EvenQ[k] && !PrimeQ[k+1] && Divisible[s3 - 2*(-1)^(k/2), k+1], AppendTo[s, k+1]]; s0 = s1; s1 = s2; s2 = s3, {k, 3, kmax}]; s]; seq[500]

PARI

lista(kmax) = {my(s0 = 1, s1 = 2, s2 = 4, s3); for(k = 3, kmax, s3 = (3*(k+1)*s2 + (k-4)*s1 - 3*(k-1)*s0)/(k+2); if(!(k % 2) && !isprime(k+1) && !((s3 - 2*(-1)^(k/2)) % (k+1)), print1(k+1, ", ")); s0 = s1; s1 = s2; s2 = s3);}