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%I #29 Mar 09 2022 00:38:09
%S 338688,826686,2239488,3188646,6613488,14224896,3416267673274176,
%T 6499837226778624
%N Numbers of multiplicative persistence 7 which are themselves the product of digits of a number.
%C 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 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 8.
%C 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 Equivalently:
%C This sequence consists of the numbers A007954(k) such that A031346(k) = 8,
%C These are the numbers k in A002473 such that A031346(k) = 7,
%C Or:
%C - they factor into powers of 2, 3, 5 and 7 exclusively.
%C - p(n) goes to a single digit in 7 steps.
%C Postulated to be finite and complete.
%C a(9), if it exists, is > 10^20000, and likely > 10^119000.
%H Daniel Mondot, <a href="https://oeis.org/wiki/File:Multiplicative_Persistence_Tree.txt">Multiplicative Persistence Tree</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/MultiplicativePersistence.html">Multiplicative Persistence</a>
%e 338688 is in this sequence because:
%e - 338688 goes to a single digit in 7 steps: p(338688) = 27648, p(27648) = 2688, p(2688)=768, p(768)=336, p(336)=54, p(54)=20, p(20)=0.
%e - p(4478976) = p(13477889) = 338688, etc.
%t mx=10^16;lst=Sort@Flatten@Table[2^i*3^j*5^k*7^l,{i,0,Log[2,mx]},{j,0,Log[3,mx/2^i]},{k,0,Log[5,mx/(2^i*3^j)]},{l,0,Log[7,mx/(2^i*3^j*5^k)]}];
%t Select[lst,Length@Most@NestWhileList[Times@@IntegerDigits@#&,#,#>9&]==7&] (* code for 7-smooth numbers from A002473. - _Giorgos Kalogeropoulos_, Jan 16 2022 *)
%Y Cf. A002473, A003001 (smallest number with multiplicative persistence n), A031346 (multiplicative persistence), A031347 (multiplicative digital root), A046516 (all numbers with mp of 7).
%Y Cf. A350180, A350181, A350182, A350183, A350184, A350185, A350187 (numbers with mp 1 to 6 and 8 to 10 that are themselves 7-smooth numbers).
%K nonn,base,more
%O 1,1
%A _Daniel Mondot_, Jan 15 2022
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