A350183 Numbers of multiplicative persistence 4 which are themselves the product of digits of a number.

378, 384, 686, 768, 1575, 1764, 2646, 4374, 6144, 6174, 6272, 7168, 8232, 8748, 16128, 21168, 23328, 27216, 28672, 32928, 34992, 49392, 59535, 67228, 77175, 96768, 112896, 139968, 148176, 163296, 214326, 236196, 393216, 642978, 691488, 774144, 777924

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 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 lists all numbers A007954(k) such that A031346(k) = 5.

- These are the numbers k in A002473 such that A031346(k) = 4.

Or:

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

- p(n) goes to a single digit in 4 steps.

Postulated to be finite and complete.

Examples

			384 is in this sequence because:
- 384 goes to a single digit in 4 steps: p(384)=96, p(96)=54, p(54)=20, p(20)=0.
- p(886)=384, p(6248)=384, p(18816)=384, etc.
378 is in this sequence because:
- 378 goes to a single digits in 4 steps: p(378)=168, p(168)=48, p(48)=32, p(32)=6.
- p(679)=378, p(2397)=378, p(12379)=378, etc.
		

Links

• Daniel Mondot, Table of n, a(n) for n = 1..142
• Eric Weisstein's World of Mathematics, Multiplicative Persistence
• Daniel Mondot, Multiplicative Persistence Tree

Crossrefs

Cf. A002473 (7-smooth), A003001 (smallest number with multiplicative persistence n), A031346 (multiplicative persistence), A031347 (multiplicative digital root), A046513 (all numbers with mp of 4).

Cf. A350180, A350181, A350182, A350184, A350185, A350186, A350187 (numbers with mp 1 to 3 and 5 to 10 that are themselves 7-smooth numbers).

Programs

Mathematica

mx=10^6;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)]}]; (* from A002473 *)
Select[lst,Length@Most@NestWhileList[Times@@IntegerDigits@#&,#,#>9&]==4&] (* Giorgos Kalogeropoulos, Jan 16 2022 *)

PARI

pd(n) = if (n, vecprod(digits(n)), 0); \\ A007954
mp(n) = my(k=n, i=0); while(#Str(k) > 1, k=pd(k); i++); i; \\ A031346
isok(k) = (mp(k)==4) && (vecmax(factor(k)[,1]) <= 7); \\ Michel Marcus, Jan 25 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 (q := pd(p)) > 9 and (r := pd(q)) > 9 and pd(r) < 10
print([k for k in range(778000) if ok(k)])