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%I #26 Sep 24 2023 10:29:42
%S 27648,47628,64827,84672,134217728,914838624,1792336896,3699376128,
%T 48814981614,134481277728,147483721728,1438916737499136
%N Numbers of multiplicative persistence 6 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 7.
%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) = 7,
%C These are the numbers k in A002473 such that A031346(k) = 6,
%C Or:
%C - they factor into powers of 2, 3, 5 and 7 exclusively.
%C - p(n) goes to a single digit in 6 steps.
%C Postulated to be finite and complete.
%C a(13), if it exists, is > 10^20000, and likely > 10^80000.
%H Daniel Mondot, <a href="https://oeis.org/wiki/File:Multiplicative_Persistence_Tree.txt">Multiplicative Persistence Tree</a>
%e 27648 is in sequence because:
%e - 27648 goes to a single digit in 6 steps: p(27648)=2688, p(2688)=768, p(768)=336, p(336)=54, p(54)=20, p(20)=0.
%e - p(338688) = p(168889) = 27648, 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&]==6&] (* code for 7-smooth numbers from A002473. - _Giorgos Kalogeropoulos_, Jan 16 2022 *)
%o (Python)
%o #this program may take 91 minutes to produce the first 8 members.
%o from math import prod
%o def hd(n):
%o while (n&1) == 0: n >>= 1
%o while (n%3) == 0: n /= 3
%o while (n%5) == 0: n /= 5
%o while (n%7) == 0: n /= 7
%o return(n)
%o def pd(n): return prod(map(int, str(n)))
%o def ok(n):
%o if hd(n) > 9: return False
%o return (p := pd(n)) > 9 and (q := pd(p)) > 9 and (r := pd(q)) > 9 and (s := pd(r)) > 9 and (t := pd(s)) > 9 and pd(t) < 10
%o print([k for k in range(10,3700000000) if ok(k)])
%Y Cf. A002473, A003001 (smallest number with multiplicative persistence n), A031346 (multiplicative persistence), A031347 (multiplicative digital root), A046515 (all numbers with mp of 6).
%Y Cf. A350180, A350181, A350182, A350183, A350184, A350186, A350187 (numbers with mp 1 to 5 and 7 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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