cp's OEIS Frontend

A350186 Numbers of multiplicative persistence 7 which are themselves the product of digits of a number.

%I A350186 #30 Feb 16 2025 08:34:02
%S A350186 338688,826686,2239488,3188646,6613488,14224896,3416267673274176,
%T A350186 6499837226778624
%N A350186 Numbers of multiplicative persistence 7 which are themselves the product of digits of a number.
%C A350186 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 A350186 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 A350186 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 A350186 Equivalently:
%C A350186 This sequence consists of the numbers A007954(k) such that A031346(k) = 8,
%C A350186 These are the numbers k in A002473 such that A031346(k) = 7,
%C A350186 Or:
%C A350186 - they factor into powers of 2, 3, 5 and 7 exclusively.
%C A350186 - p(n) goes to a single digit in 7 steps.
%C A350186 Postulated to be finite and complete.
%C A350186 a(9), if it exists, is > 10^20000, and likely > 10^119000.
%H A350186 Daniel Mondot, <a href="https://oeis.org/wiki/File:Multiplicative_Persistence_Tree.txt">Multiplicative Persistence Tree</a>
%H A350186 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/MultiplicativePersistence.html">Multiplicative Persistence</a>
%e A350186 338688 is in this sequence because:
%e A350186 - 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 A350186 - p(4478976) = p(13477889) = 338688, etc.
%t A350186 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 A350186 Select[lst,Length@Most@NestWhileList[Times@@IntegerDigits@#&,#,#>9&]==7&]  (* code for 7-smooth numbers from A002473. - _Giorgos Kalogeropoulos_, Jan 16 2022 *)
%Y A350186 Cf. A002473, A003001 (smallest number with multiplicative persistence n), A031346 (multiplicative persistence), A031347 (multiplicative digital root), A046516 (all numbers with mp of 7).
%Y A350186 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 A350186 nonn,base,more
%O A350186 1,1
%A A350186 _Daniel Mondot_, Jan 15 2022