A350185 Numbers of multiplicative persistence 6 which are themselves the product of digits of a number.

27648, 47628, 64827, 84672, 134217728, 914838624, 1792336896, 3699376128, 48814981614, 134481277728, 147483721728, 1438916737499136

Offset: 1

Author

Daniel Mondot, Jan 15 2022

Keywords

• nonn
• base
• more

Comments

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).

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.

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....

Equivalently:

This sequence consists of the numbers A007954(k) such that A031346(k) = 7,

These are the numbers k in A002473 such that A031346(k) = 6,

Or:

- they factor into powers of 2, 3, 5 and 7 exclusively.

- p(n) goes to a single digit in 6 steps.

Postulated to be finite and complete.

a(13), if it exists, is > 10^20000, and likely > 10^80000.

Examples

			27648 is in sequence because:
- 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.
- p(338688) = p(168889) = 27648, etc.
		

Links

• Daniel Mondot, Multiplicative Persistence Tree

Crossrefs

Cf. A002473, A003001 (smallest number with multiplicative persistence n), A031346 (multiplicative persistence), A031347 (multiplicative digital root), A046515 (all numbers with mp of 6).

Cf. A350180, A350181, A350182, A350183, A350184, A350186, A350187 (numbers with mp 1 to 5 and 7 to 10 that are themselves 7-smooth numbers).

Programs

Mathematica

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)]}];
Select[lst,Length@Most@NestWhileList[Times@@IntegerDigits@#&,#,#>9&]==6&] (* code for 7-smooth numbers from A002473. - Giorgos Kalogeropoulos, Jan 16 2022 *)

Python

#this program may take 91 minutes to produce the first 8 members.
from math import prod
def hd(n):
    while (n&1) == 0:  n >>= 1
    while (n%3) == 0:  n /= 3
    while (n%5) == 0:  n /= 5
    while (n%7) == 0:  n /= 7
    return(n)
def pd(n): return prod(map(int, str(n)))
def ok(n):
    if hd(n) > 9: return False
    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
print([k for k in range(10,3700000000) if ok(k)])