%I #27 Mar 09 2022 10:52:41
%S 1,2,3,5,8,10,11,12,13,15,18,20,21,24,25,27,30,31,42,45,50,51,52,54,
%T 55,56,57,65,72,75,80,81,100,101,102,103,105,108,110,111,112,113,115,
%U 118,120,121,124,125,127,130,131,142,145,150,151,152,154,155,156,157,165,172,175,180,181
%N Numbers whose trajectory under iteration of the product of cubes of nonzero digits map includes 1 (conjectured).
%C 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.
%C Every power 10^k is a term of this sequence.
%C If k is a term, the numbers obtained by inserting zeros anywhere in k are terms.
%C If k is a term, the numbers obtained by inserting ones anywhere in k are terms.
%C If k is a term, each distinct permutation of the digits of k gives another term.
%C If k is a term, the number of iterations required to converge to 1 is less than or equal to 10 (conjectured).
%e 217 is a term of the sequence; its trajectory is 217 > 2744 > 11239424 > 5159780352 > 54010152000000000 > 8000000 > 512 > 1000 > 1.
%e 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.
%t Select[Range[1000], FixedPoint[ Product[ReplaceAll[0 > 1][IntegerDigits[#]][[i]]^3, {i, 1, Length[ReplaceAll[0 > 1][IntegerDigits[#]]]}] &, #, 12] == 1 &]
%Y Cf. A351327, A051801.
%Y Cf. A352172 (product of cubes of nonzero digits).
%K nonn,base
%O 1,2
%A _Luca Onnis_, Feb 23 2022
