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A350181
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Numbers of multiplicative persistence 2 which are themselves the product of digits of a number.
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8
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25, 27, 28, 35, 36, 45, 48, 54, 56, 63, 64, 72, 84, 125, 126, 128, 135, 144, 162, 192, 216, 224, 225, 243, 245, 252, 256, 315, 324, 375, 432, 441, 512, 525, 567, 576, 588, 625, 675, 735, 756, 875, 945, 1125, 1134, 1152, 1176, 1215, 1225, 1296, 1323, 1372
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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 3.
There are infinitely many numbers with mp of 1 to 11, but the classes of numbers (p(n)) are postulated to be finite for this and subsequent sequences A350182....
Equivalently:
This sequence consists of the numbers A007954(k) such that A031346(k) = 3,
Or:
- they factor into powers of 2, 3, 5 and 7 exclusively.
- p(n) goes to a single digit in 2 steps.
Postulated to be finite and complete.
The largest known number is 2^25 * 3^227 * 7^28 (140 digits).
No more numbers have been found between 10^140 and probably 10^20000 (according to comment in A003001), and independently verified up to 10^10000.
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LINKS
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EXAMPLE
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25 is in this sequence because:
- 25 goes to a single digit in 2 steps: p(25) = 10, p(10) = 0.
- 25 has ancestors 55, 155, etc. p(55) = 25.
27 is in this sequence because:
- 27 goes to a single digit in 2 steps: p(27) = 14, p(14) = 4.
- 27 has ancestors 39, 93, 333, 139, etc. p(39) = 27.
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MATHEMATICA
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Select[Range@1400, AllTrue[First/@FactorInteger@#, #<10&]&&Length@Most@NestWhileList[Times@@IntegerDigits@#&, #, #>9&]==2&] (* Giorgos Kalogeropoulos, Jan 16 2022 *)
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PROG
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(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 pd(p) < 10
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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), A046511 (all numbers with mp of 2).
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KEYWORD
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base,nonn
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AUTHOR
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STATUS
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approved
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