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:
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
KEYWORD
nonn,base
AUTHOR
Daniel Mondot, Dec 18 2021
STATUS
approved