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

2688, 18816, 26244, 98784, 222264, 262144, 331776, 333396, 666792, 688128, 1769472, 2939328, 3687936, 4214784, 4917248, 13226976, 19361664, 38118276, 71663616, 111476736, 133413966, 161414428, 169869312, 184473632, 267846264, 368947264, 476171136, 1783627776

Offset

1,1

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 5.

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 all numbers A007954(k) such that A031346(k) = 6.

These are the numbers k in A002473 such that A031346(k) = 5.

Or:

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

- p(n) goes to a single digit in 5 steps.

Postulated to be finite and complete.

Links

Daniel Mondot, Table of n, a(n) for n = 1..41

Daniel Mondot, Multiplicative Persistence Tree

Eric Weisstein's World of Mathematics, Multiplicative Persistence

Example

2688 is in this sequence because:
- 2688 goes to a single digit in 5 steps: p(2688)=768, p(768)=336, p(336)=54, p(54)=20, p(20)=0.
- p(27648) = p(47628) = 2688, etc.
331776 is in this sequence because:
- 331776 goes to a single digit in 5 steps: p(331776)=2646, p(2646)=288, p(288)=128, p(128)=16, p(16)=6.
- p(914838624) = p(888899) = 331776, etc.

Mathematica

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)]}];
Select[lst, Length@Most@NestWhileList[Times@@IntegerDigits@#&, #, #>9&]==5&] (* code for 7-smooth numbers from A002473. - Giorgos Kalogeropoulos, Jan 16 2022 *)

Prog

(Python)
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 pd(s) < 10
print([k for k in range(10, 476200000) if ok(k)])

Crossrefs

Intersection of A002473 and A046514 (all numbers with mp of 5).

Cf. A003001 (smallest number with multiplicative persistence n), A031346 (multiplicative persistence), A031347 (multiplicative digital root).

Cf. A350180, A350181, A350182, A350183, A350185, A350186, A350187 (numbers with mp 1 to 4 and 6 to 10 that are themselves 7-smooth numbers).

Sequence in context: A243993 A157613 A233899 * A186833 A229682 A186937

Adjacent sequences: A350181 A350182 A350183 * A350185 A350186 A350187

Keyword

nonn,base

Author

Daniel Mondot, Dec 18 2021

Status

approved