login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A350184 Numbers of multiplicative persistence 5 which are themselves the product of digits of a number. 8
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 (list; graph; refs; listen; history; text; internal format)
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
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
KEYWORD
nonn,base
AUTHOR
Daniel Mondot, Dec 18 2021
STATUS
approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified April 13 04:33 EDT 2024. Contains 371639 sequences. (Running on oeis4.)