A350181 Numbers of multiplicative persistence 2 which are themselves the product of digits of a number.

25, 27, 28, 35, 36, 45, 48, 54, 56, 63, 64, 72, 84, 125, 126, 128, 135, 144, 162, 192, 216, 224, 225, 243, 245, 252, 256, 315, 324, 375, 432, 441, 512, 525, 567, 576, 588, 625, 675, 735, 756, 875, 945, 1125, 1134, 1152, 1176, 1215, 1225, 1296, 1323, 1372

Offset: 1

Author

Daniel Mondot, Dec 18 2021

Keywords

• base
• nonn

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 3.

There are infinitely many numbers with mp of 1 to 11, but the classes of numbers (p(n)) are postulated to be finite for this and subsequent sequences A350182....

Equivalently:

This sequence consists of the numbers A007954(k) such that A031346(k) = 3,

These are the numbers k in A002473 such that A031346(k) = 2,

Or:

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

- p(n) goes to a single digit in 2 steps.

Postulated to be finite and complete.

The largest known number is 2^25 * 3^227 * 7^28 (140 digits).

No more numbers have been found between 10^140 and probably 10^20000 (according to comment in A003001), and independently verified up to 10^10000.

Examples

			25 is in this sequence because:
- 25 goes to a single digit in 2 steps: p(25) = 10, p(10) = 0.
- 25 has ancestors 55, 155, etc. p(55) = 25.
27 is in this sequence because:
- 27 goes to a single digit in 2 steps: p(27) = 14, p(14) = 4.
- 27 has ancestors 39, 93, 333, 139, etc. p(39) = 27.
		

Links

• Daniel Mondot, Table of n, a(n) for n = 1..11994

Crossrefs

Cf. A002473, A003001 (smallest number with multiplicative persistence n), A031346 (multiplicative persistence), A031347 (multiplicative digital root), A046511 (all numbers with mp of 2).

Cf. A350180, A350182, A350183, A350184, A350185, A350186, A350187 (numbers with mp 1, and 3 to 10 that are themselves 7-smooth numbers).

Programs

Mathematica

Select[Range@1400,AllTrue[First/@FactorInteger@#,#<10&]&&Length@Most@NestWhileList[Times@@IntegerDigits@#&,#,#>9&]==2&] (* Giorgos Kalogeropoulos, Jan 16 2022 *)

Python

from math import prod
from sympy import factorint
def pd(n): return prod(map(int, str(n)))
def ok(n):
    if n <= 9 or max(factorint(n)) > 9: return False
    return (p := pd(n)) > 9 and pd(p) < 10
print([k for k in range(1400) if ok(k)]) # Michael S. Branicky, Jan 16 2022