A396944 a(n) is the smallest integer k such that the iterated multiplicative-order chain k -> ord_k(2) -> ord_{ord_k(2)}(2) -> ... reaches an even number in exactly n steps.

2, 3, 7, 47, 1439, 2879, 214559, 429119, 858239, 164255999

Offset

0,1

Comments

The chain is formed by repeating the following: for any integer q, calculate ord_q(2) (the multiplicative order of 2 modulo q), then continue this process. The chain stops when it reaches an even number.

Links

Table of n, a(n) for n=0..9.

Example

a(1): 3 -> 2.
a(2): 7 -> 3 -> 2.
a(3): 47 -> 23 -> 11 -> 10.
a(4): 1439 -> 719 -> 359 -> 179 -> 178.
a(5): 2879 -> 1439 -> 719 -> 359 -> 179 -> 178.
a(6): 214559 -> 107279 -> 53639 -> 2063 -> 1031 -> 515 -> 204.
a(7): 429119 -> 214559 -> 107279 -> 53639 -> 2063 -> 1031 -> 515 -> 204.
a(8): 858239 -> 429119 -> 214559 -> 107279 -> 53639 -> 2063 -> 1031 -> 515 -> 204.

Prog

(PARI) isok(k, n) = for (i=1, n-1, k = znorder(Mod(2, k)); if (!(k%2), return(0))); k = znorder(Mod(2, k)); if (k % 2, 0, 1);
a(n) = my(k=1); while(!isok(k, n), k+=2); k; \\ Michel Marcus, Jun 11 2026
(Python)
from sympy import n_order
def steps(k):
    q, s = k, 0
    while q % 2 == 1:
        q = n_order(2, q)
        s += 1
    return s
def a(n):
    k = 2
    while steps(k) != n:
        k += 1
    return k
print([a(n) for n in range(6)])

Crossrefs

Cf. A005384, A059452, A002326.

Sequence in context: A257108 A274385 A075461 * A059785 A271041 A270402

Adjacent sequences: A396941 A396942 A396943 * A396946 A396948 A396949

Keyword

nonn,hard,more,new

Author

Nishant R. Gautam, Jun 11 2026

Extensions

a(9) from Michel Marcus, Jun 11 2026

Status

approved