cp's OEIS Frontend

A350182 Numbers of multiplicative persistence 3 which are themselves the product of digits of a number.

%I A350182 #27 Feb 16 2025 08:34:02
%S A350182 49,75,96,98,147,168,175,189,196,288,294,336,343,392,448,486,648,672,
%T A350182 729,784,864,882,896,972,1344,1715,1792,1944,2268,2744,3136,3375,3888,
%U A350182 3969,7938,8192,9375,11664,12288,12348,13824,14336,16384,16464,17496,18144
%N A350182 Numbers of multiplicative persistence 3 which are themselves the product of digits of a number.
%C A350182 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 A350182 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 4.
%C A350182 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 A350182 Equivalently:
%C A350182 This sequence consists of the numbers A007954(k) such that A031346(k) = 4,
%C A350182 These are the numbers k in A002473 such that A031346(k) = 3,
%C A350182 Or:
%C A350182 - they factor into powers of 2, 3, 5 and 7 exclusively.
%C A350182 - p(n) goes to a single digit in 3 steps.
%C A350182 Postulated to be finite and complete.
%C A350182 Let p(n) be the product of all the digits of n.
%C A350182 The multiplicative persistence of a number mp(n) is the number of times you need to apply p() to get to a single digit.
%C A350182 For example:
%C A350182 mp(1) is 0 since 1 is already a single-digit number.
%C A350182 mp(10) is 1 since p(10) = 0, and 0 is a single digit, 1 step.
%C A350182 mp(25) is 2 since p(25) = 10, p(10) = 0, 2 steps.
%C A350182 mp(96) is 3 since p(96) = 54, p(54) = 20, p(20) = 0, 3 steps.
%C A350182 mp(378) is 4 since p(378) = 168, p(168) = 48, p(48) = 32, p(32) = 6, 4 steps.
%C A350182 There are infinitely many numbers n such that mp(n)=4. But for each n with mp(n)=4, p(n) is a number included in this sequence, and this sequence is likely finite.
%C A350182 This sequence lists p(n) such that mp(n) = 4, or mp(p(n)) = 3.
%H A350182 Daniel Mondot, <a href="/A350182/b350182.txt">Table of n, a(n) for n = 1..387</a>
%H A350182 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/MultiplicativePersistence.html">Multiplicative Persistence</a>
%H A350182 <a href="https://oeis.org/wiki/File:Multiplicative_Persistence_Tree.txt">Multiplicative Persistence Tree</a>
%e A350182 49 is in this sequence because:
%e A350182 - 49 goes to a single digit in 3 steps: p(49) = 36, p(36) = 18, p(18) = 8.
%e A350182 - p(77) = p(177) = p(717) = p(771) = 49, etc.
%e A350182 75 is in this sequence because:
%e A350182 - 75 goes to a single digit in 3 steps: p(75) = 35, p(35) = 15, p(15) = 5.
%e A350182 - p(355) = p(535) = p(1553) = 75, etc.
%Y A350182 Cf. A002473, A003001 (smallest number with multiplicative persistence n), A031346 (multiplicative persistence), A031347 (multiplicative digital root), A046512 (all numbers with mp of 3).
%Y A350182 Cf. A350180, A350181, A350183, A350184, A350185, A350186, A350187 (numbers with mp 0, 1 and 3 to 10 that are themselves 7-smooth numbers).
%K A350182 base,nonn
%O A350182 1,1
%A A350182 _Daniel Mondot_, Dec 18 2021