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A350185
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Numbers of multiplicative persistence 6 which are themselves the product of digits of a number.
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8
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27648, 47628, 64827, 84672, 134217728, 914838624, 1792336896, 3699376128, 48814981614, 134481277728, 147483721728, 1438916737499136
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OFFSET
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1,1
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COMMENTS
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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:
This sequence consists of the numbers A007954(k) such that A031346(k) = 7,
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.
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LINKS
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EXAMPLE
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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.
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MATHEMATICA
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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 *)
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PROG
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(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)])
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CROSSREFS
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Cf. A002473, A003001 (smallest number with multiplicative persistence n), A031346 (multiplicative persistence), A031347 (multiplicative digital root), A046515 (all numbers with mp of 6).
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KEYWORD
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nonn,base,more
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AUTHOR
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STATUS
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approved
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