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 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:
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.
LINKS
Daniel Mondot, Multiplicative Persistence Tree
EXAMPLE
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.
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 *)
PROG
(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)])
CROSSREFS
Cf. A002473, A003001 (smallest number with multiplicative persistence n), A031346 (multiplicative persistence), A031347 (multiplicative digital root), A046515 (all numbers with mp of 6).
KEYWORD
nonn,base,more
AUTHOR
Daniel Mondot, Jan 15 2022
STATUS
approved