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A350180 Numbers of multiplicative persistence 1 which are themselves the product of digits of a number.

%I A350180 #59 Feb 16 2025 08:34:02
%S A350180 10,12,14,15,16,18,20,21,24,30,32,40,42,50,60,70,80,81,90,100,105,108,
%T A350180 112,120,140,150,160,180,200,210,240,250,270,280,300,320,350,360,400,
%U A350180 405,420,450,480,490,500,504,540,560,600,630,640,700,720,750,800
%N A350180 Numbers of multiplicative persistence 1 which are themselves the product of digits of a number.
%C A350180 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 A350180 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 2.
%C A350180 There are infinitely many numbers with mp of 1 to 11, but the classes of numbers (p(n)) are postulated to be finite for subsequent sequences A350181..., but not for this sequence (where mp(p(n)) = 1). That is because there are infinitely many numbers that include both an even digit (2, 4, 6 or 8), a 5 and no 0. For these numbers n, p(n) will include a zero and p(p(n)) will be 0.
%C A350180 Equivalently: This sequence contains all numbers A007954(k) such that A031346(k) = 2, and they are the numbers k in A002473 such that A031346(k) = 1.
%C A350180 Or, they factor into powers of 2, 3, 5 and 7 exclusively and p(n) goes to a single digit in 1 step.
%H A350180 Daniel Mondot, <a href="/A350180/b350180.txt">Table of n, a(n) for n = 1..20000</a>
%H A350180 Daniel Mondot, <a href="https://oeis.org/wiki/File:Multiplicative_Persistence_Tree.txt">Multiplicative Persistence Tree</a>
%H A350180 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/MultiplicativePersistence.html">Multiplicative Persistence</a>
%e A350180 10 is in this sequence because:
%e A350180 - 10 goes to a single digit in 1 step: p(10) = 0.
%e A350180 - 25, 52, 125, 152, 215, 512, 251, 521, 1125, 1152, 1215, 1512, 1251, 1521, 2115, 5112, 2511, 5211, etc. all lead to 10, i.e., p(25)=10, p(52)=10, etc.
%e A350180 Some of these (25, 125, 512, 1125, 1152, 1215, 1512) are in the next layer of classes, A350181, and the rest are not.
%e A350180 12 is in this sequence because:
%e A350180 - 12 goes to a single digit in 1 step: p(12) = 2.
%e A350180 - 12, 21, 112, 211, 121, 11112, 11211, etc. all lead to 12.
%e A350180 (12, 21 and 112 are in the next layer of classes, A350181, but the rest are not)
%e A350180 14 is in this sequence because:
%e A350180 - 14 goes to a single digit in 1 step: p(14) = 4.
%e A350180 - 27, 72, 127, 172, 217, 712, 271, 721, 12111711, etc. all lead to 14.
%e A350180 (27 and 72 are in the next layer of classes, A350181, the rest are not).
%o A350180 (PARI)
%o A350180 mp(n)={my(k=0); while(n>=10, k++; n=vecprod(digits(n))); k}
%o A350180 isparent(n)={my(m=0); while(m<>n, m=n; n/=gcd(n,2*3*5*7)); n==1}
%o A350180 isok(n)={mp(n)==1 && isparent(n)} \\ _Andrew Howroyd_, Dec 20 2021
%Y A350180 Intersection of A002473 and A046510
%Y A350180 Cf. A003001 (smallest number with multiplicative persistence n), A031346 (multiplicative persistence), A031347 (multiplicative digital root), A046510 (all numbers with mp of 1).
%Y A350180 Cf. A350181, A350182, A350183, A350184, A350185, A350186, A350187 (numbers with mp 2 to 10 that are themselves 7-smooth numbers).
%K A350180 base,nonn
%O A350180 1,1
%A A350180 _Daniel Mondot_, Dec 18 2021