A351327 -id:A351327 - cp's OEIS frontend

A351876 Numbers whose trajectory under iteration of the product of cubes of nonzero digits map includes 1 (conjectured).

1, 2, 3, 5, 8, 10, 11, 12, 13, 15, 18, 20, 21, 24, 25, 27, 30, 31, 42, 45, 50, 51, 52, 54, 55, 56, 57, 65, 72, 75, 80, 81, 100, 101, 102, 103, 105, 108, 110, 111, 112, 113, 115, 118, 120, 121, 124, 125, 127, 130, 131, 142, 145, 150, 151, 152, 154, 155, 156, 157, 165, 172, 175, 180, 181

Offset: 1

Luca Onnis, Feb 23 2022

To determine whether a given number k is a term of this sequence, start with k, take the cube of the product of its nonzero digits, apply the same process to the result, and continue until 30 iterations are reached. If 1 is reached during the process, k is a term of this sequence. If not, k is not a term of this sequence.
Every power 10^k is a term of this sequence.
If k is a term, the numbers obtained by inserting zeros anywhere in k are terms.
If k is a term, the numbers obtained by inserting ones anywhere in k are terms.
If k is a term, each distinct permutation of the digits of k gives another term.
If k is a term, the number of iterations required to converge to 1 is less than or equal to 10 (conjectured).

			217 is a term of the sequence; its trajectory is 217 -> 2744 -> 11239424 -> 5159780352 -> 54010152000000000 -> 8000000 -> 512 -> 1000 -> 1.
4 is not a term of the sequence; its trajectory begins with 4 -> 64 -> 13824 -> 7077888 -> 5416169448144896 -> 188436971148778297205194752000 -> 1545896640285238037724131582088286996267008000000 -> ... Subsequent terms in the trajectory get larger and larger, rather than reaching 1. However, it is not yet known whether it eventually reaches 1 after some number of iterations > 30.

Cf. A351327, A051801.

Cf. A352172 (product of cubes of nonzero digits).

Select[Range[1000], FixedPoint[ Product[ReplaceAll[0 -> 1][IntegerDigits[#]][[i]]^3, {i, 1, Length[ReplaceAll[0 -> 1][IntegerDigits[#]]]}] &, #, 12] == 1 &]

A352595 Positive integers that are fixed points for the map x->f^k(x) for some k>1, where f(x) is the product of squares of nonzero digits of x.

Original entry on oeis.org

256, 324, 576, 3600, 11664, 15876, 20736, 44100, 63504, 65536, 129600, 2822400, 5308416, 7290000, 8294400

Offset: 1

Michael S. Branicky, Mar 24 2022

f(x) = A352598(x).
Fixed points of f(x) are in A115385.
64524128256, 386983526400, 849346560000, 49787136000000, 55725627801600 are also terms.

			256 -> 3600 -> 324 -> 576 -> 44100 -> 256 is a limit cycle of f, so all elements are terms.

Subsequence of the intersection of A000290 and A002473.

Cf. A115385, A351327, A352598.

from math import prod
from itertools import count, islice
def f(n): return prod(int(d)**2 for d in str(n) if d != "0")
def ok(n):
    n0, k, seen = n, 0, set(),
    while n not in seen: # iterate until fixed point or in cycle
        seen.add(n)
        n = f(n)
        k += 1
    return n == n0 and k > 1
def agen(startk=1):
    for m in count(1):
        if ok(m): yield m
print(list(islice(agen(), 11)))

cp's OEIS Frontend

A351876 Numbers whose trajectory under iteration of the product of cubes of nonzero digits map includes 1 (conjectured).

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica