cp's OEIS Frontend

A350187 Numbers of multiplicative persistence 8 which are themselves the product of digits of a number.

%I A350187 #21 Feb 16 2025 08:34:02
%S A350187 4478976,784147392,19421724672,249143169618,717233481216
%N A350187 Numbers of multiplicative persistence 8 which are themselves the product of digits of a number.
%C A350187 The multiplicative persistence of a number mp(n) is the number of times the product of digits function p(n) must be applied to reach a single digit, i.e., A031346(n).
%C A350187 The product of digits function partitions all numbers into equivalence classes. There is a one-to-one correspondence between values in this sequence and equivalence classes of numbers with multiplicative persistence 9.
%C A350187 There are infinitely many numbers with mp of 1 to 11, but the classes of numbers (p(n)) are postulated to be finite for sequences A350181.
%C A350187 Equivalently:
%C A350187 This sequence consists of all numbers A007954(k) such that A031346(k) = 9.
%C A350187 They are the numbers k in A002473 such that A031346(k) = 8.
%C A350187 Or they factor into powers of 2, 3, 5 and 7 exclusively and p(n) goes to a single digit in 8 steps.
%C A350187 Postulated to be finite and complete.
%C A350187 a(6), if it exists, is > 10^20000, and likely > 10^171000.
%H A350187 Daniel Mondot, <a href="https://oeis.org/wiki/File:Multiplicative_Persistence_Tree.txt">Multiplicative Persistence Tree</a>
%H A350187 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/MultiplicativePersistence.html">Multiplicative Persistence</a>
%e A350187 4478976 is in this sequence because:
%e A350187 - 4478976 goes to a single digit in 8 steps: 4478976 -> 338688 -> 27648 -> 2688 -> 768 -> 336 -> 54 -> 20 -> 0;
%e A350187 - p(438939648) = p(231928233984) = 4478976.
%Y A350187 Intersection of A002473 and A046517.
%Y A350187 Cf. A003001 (smallest number with multiplicative persistence n), A031346 (multiplicative persistence), A031347 (multiplicative digital root), A046517 (all numbers with mp of 8).
%Y A350187 Cf. A350180, A350181, A350182, A350183, A350184, A350185, A350186 (numbers with mp 1 to 7 and 9 to 10 that are themselves 7-smooth numbers).
%K A350187 nonn,base,more
%O A350187 1,1
%A A350187 _Daniel Mondot_, Jan 30 2022