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A350184 Numbers of multiplicative persistence 5 which are themselves the product of digits of a number.

%I A350184 #39 Feb 16 2025 08:34:02
%S A350184 2688,18816,26244,98784,222264,262144,331776,333396,666792,688128,
%T A350184 1769472,2939328,3687936,4214784,4917248,13226976,19361664,38118276,
%U A350184 71663616,111476736,133413966,161414428,169869312,184473632,267846264,368947264,476171136,1783627776
%N A350184 Numbers of multiplicative persistence 5 which are themselves the product of digits of a number.
%C A350184 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 A350184 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 5.
%C A350184 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 A350184 Equivalently:
%C A350184 This sequence consists of all numbers A007954(k) such that A031346(k) = 6.
%C A350184 These are the numbers k in A002473 such that A031346(k) = 5.
%C A350184 Or:
%C A350184 - they factor into powers of 2, 3, 5 and 7 exclusively.
%C A350184 - p(n) goes to a single digit in 5 steps.
%C A350184 Postulated to be finite and complete.
%H A350184 Daniel Mondot, <a href="/A350184/b350184.txt">Table of n, a(n) for n = 1..41</a>
%H A350184 Daniel Mondot, <a href="https://oeis.org/wiki/File:Multiplicative_Persistence_Tree.txt">Multiplicative Persistence Tree</a>
%H A350184 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/MultiplicativePersistence.html">Multiplicative Persistence</a>
%e A350184 2688 is in this sequence because:
%e A350184 - 2688 goes to a single digit in 5 steps: p(2688)=768, p(768)=336, p(336)=54, p(54)=20, p(20)=0.
%e A350184 - p(27648) = p(47628) = 2688, etc.
%e A350184 331776 is in this sequence because:
%e A350184 - 331776 goes to a single digit in 5 steps: p(331776)=2646, p(2646)=288, p(288)=128, p(128)=16, p(16)=6.
%e A350184 - p(914838624) = p(888899) = 331776, etc.
%t A350184 mx=10^10;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 A350184 Select[lst,Length@Most@NestWhileList[Times@@IntegerDigits@#&,#,#>9&]==5&] (* code for 7-smooth numbers from A002473. - _Giorgos Kalogeropoulos_, Jan 16 2022 *)
%o A350184 (Python)
%o A350184 from math import prod
%o A350184 def hd(n):
%o A350184     while (n&1) == 0:  n >>= 1
%o A350184     while (n%3) == 0:  n /= 3
%o A350184     while (n%5) == 0:  n /= 5
%o A350184     while (n%7) == 0:  n /= 7
%o A350184     return(n)
%o A350184 def pd(n): return prod(map(int, str(n)))
%o A350184 def ok(n):
%o A350184     if hd(n) > 9: return False
%o A350184     return (p := pd(n)) > 9 and (q := pd(p)) > 9 and (r := pd(q)) > 9 and (s := pd(r)) > 9 and pd(s) < 10
%o A350184 print([k for k in range(10,476200000) if ok(k)])
%Y A350184 Intersection of A002473 and A046514 (all numbers with mp of 5).
%Y A350184 Cf. A003001 (smallest number with multiplicative persistence n), A031346 (multiplicative persistence), A031347 (multiplicative digital root).
%Y A350184 Cf. A350180, A350181, A350182, A350183, A350185, A350186, A350187 (numbers with mp 1 to 4 and 6 to 10 that are themselves 7-smooth numbers).
%K A350184 nonn,base
%O A350184 1,1
%A A350184 _Daniel Mondot_, Dec 18 2021