A393215 Odd composite numbers k such that M(k-3) == (-3)^((k-3)/2) (mod k), where M(k) = A001006(k) is the k-th Motzkin number.

121, 171, 891, 1001, 1701, 11907, 13851, 16767, 32805, 82377, 98415, 191241, 373977, 511029, 859491, 885735, 2355075, 3182085, 3956283, 4074381, 6008067, 12223143, 12987521, 15483311, 21552885, 25135191, 31040091, 37732311, 44625735, 59422977, 70694775, 89579925

Offset

1,1

Comments

The congruence holds for every odd prime (Kohen, 2026).

Links

Chai Wah Wu, Table of n, a(n) for n = 1..35

Nadav Kohen, Density and Symmetry in the Generalized Motzkin Numbers Modulo p, INTEGERS 26 (2026), #A22.

Example

121 = 11^2 is a term since it is an odd composite number, and M(121-3) - (-3)^((121-3)/2) = 121 * 1848734059536540155671460409071712328042981037485570 is a multiple of 121.

Mathematica

seq[kmax_] := Module[{s = {}, mot1 = 1, mot2 = 2, mot}, Do[mot3 = ((2*k + 1)*mot2 + (3*k - 3)*mot1)/(k + 2); If[EvenQ[k] && CompositeQ[k + 3] && Divisible[mot3 - (-3)^(k/2), k + 3], AppendTo[s, k + 3]]; mot1 = mot2; mot2 = mot3, {k, 3, kmax}]; s]; seq[10^5]

Prog

(PARI) list(kmax) = {my(mot1 = 1, mot2 = 2, mot); for(k = 3, kmax, mot3 = ((2*k+1)*mot2 + (3*k-3)*mot1)/(k+2); if(!(k % 2) && !isprime(k+3) && !((mot3-(-3)^(k/2)) % (k+3)), print1(k+3, ", ")); mot1 = mot2; mot2 = mot3); }

Crossrefs

Cf. A001006, A081735, A081736.

Sequence in context: A346316 A284643 A074730 * A268519 A364778 A037050

Adjacent sequences: A393212 A393213 A393214 * A393216 A393217 A393218

Keyword

nonn

Author

Amiram Eldar, Feb 06 2026

Extensions

a(27)-a(28) from Chai Wah Wu, Feb 10 2026

a(29)-a(32) from Chai Wah Wu, Feb 19 2026

Status

approved