A349249 Number of iterations of the digit-census function A348783 until an element of a cycle is reached.

6, 5, 3, 4, 6, 8, 10, 12, 14, 16, 4, 3, 1, 2, 4, 6, 8, 10, 12, 14, 2, 0, 0, 2, 4, 6, 8, 10, 12, 14, 0, 2, 2, 6, 4, 6, 8, 10, 12, 14, 4, 4, 4, 4, 6, 6, 8, 10, 12, 14, 6, 6, 6, 6, 6, 10, 8, 10, 12, 14, 8, 8, 8, 8, 8, 8, 8, 10, 12, 14, 10, 10, 10, 10, 10, 10, 10

Offset

0,1

Comments

The length of the cycle in which the trajectory of n will end, is equal to the size of the full orbit of n under A348783, A349250(n), minus a(n).

Links

Table of n, a(n) for n=0..76.

Formula

a(n) = 0 iff n is element of a cycle (or "loop") under iterations of A348783,

otherwise a(n) = 1 + a(A348783(n)).

Example

Under iterations of A348783, 0 -> 1 -> 10 -> 11 -> 20 -> 101 -> 21 -> 110 -> 21 -> ..., so the cycle (21, 110) is reached after a(0) = 6 iterations, and also a(1) = 5, a(10) = 4, a(11) = 3, a(20) = 2, a(101) = 1 and a(21) = a(110) = 0 because these two are elements of this cycle.
Similarly, 2 -> 100 -> 12 -> 110 -> 21 -> ..., so the same cycle is reached in a(2) = 3 iterations, and also a(100) = 2, a(12) = 1.
Then, 3 -> 1000 -> 13 -> 1010 -> 22 -> 200 -> 102 -> 111 -> 30 -> 1001 -> 22 -> 200 -> ..., here the cycle (22, 200, 102, 111, 30, 1001) is reached after a(3) = 4 iterations, and a(1000) = 3, a(13) = 2, a(1010) = 1 and a(n) = 0 for the elements of that cycle.

Prog

(PARI) apply( {A349249(n, S=[n], T)=while(!setsearch(S, n=A348783(n)), S=setunion(S, [n])); if(T, T-#S, A349249(n, , #S))}, [0..99])
(Python)
def f(n):
    s = str(n)
    return int("".join(str(s.count(d)) for d in "9876543210").lstrip("0"))
def a(n):
    orbit, fn = [n], f(n)
    while fn not in orbit:
        orbit.append(fn)
        n, fn = fn, f(fn)
    return orbit.index(fn)
print([a(n) for n in range(77)]) # Michael S. Branicky, Nov 18 2021

Crossrefs

Cf. A348783, A349250 (length of the full orbit).

Sequence in context: A225662 A225663 A225664 * A225665 A019686 A143304

Adjacent sequences: A349246 A349247 A349248 * A349250 A349251 A349252

Keyword

nonn

Author

M. F. Hasler, Nov 17 2021

Status

approved