A066736 Least number > 1 such that the nonzero product of the digits of its n-th power is also an n-th power.

2, 2, 2, 118, 382, 580408, 178758, 12, 1254514, 53067715, 51773525, 314537797, 110242999

Offset

1,1

Comments

The n-th roots of the product of the digits of the n-th power of a(n) are 2, 2, 2, 42, 24, 1440, 2520, 6, 10080. Because the numbers get larger quicker, the available candidates decreases. See A020665. Therefore this sequence might be finite or have a preponderance of blank entries.

If a(14) exists, it is greater than 4*10^8. - Derek Orr, Feb 17 2014

If a(14) exists, it is greater than 5.6*10^10. - Sean A. Irvine, Nov 05 2023

Links

Table of n, a(n) for n=1..13.

Example

382^5 = 8134236862432 and the product of these digits is 7962624 = 24^5 (another fifth power). Since 382 is the smallest number with this property, a(5) = 382.

Mathematica

Do[k = 2; While[a = Apply[Times, IntegerDigits[k^n]]; a == 0 || !IntegerQ[a^(1/n)], k++ ]; Print[k], {n, 1, 10} ]

Prog

(Python)
import sympy
from sympy import factorint
def DigitProd(x):
..total = 1
..for i in str(x):
....total *= int(i)
..return total
def Prod(x):
..n = 2
..while n < 4*(10**8):
....if DigitProd(n**x) != 0 and DigitProd(n**x) != 1:
......count = 0
......for i in list(factorint(DigitProd(n**x)).values()):
........if (int(i)/x) % 1 == 0:
..........count += 1
........else:
..........break
......if count == len(list(factorint(DigitProd(n**x)).values())):
........return n
......else:
........n += 1
....else:
......n += 1
x = 1
while x < 100:
..print(Prod(x))
..x += 1 # Derek Orr, Feb 13 2014

Crossrefs

Sequence in context: A102384 A275533 A304463 * A324302 A216668 A057330

Adjacent sequences: A066733 A066734 A066735 * A066737 A066738 A066739

Keyword

nonn,base

Author

Robert G. Wilson v, Jan 15 2002

Extensions

a(10)-a(13) from Derek Orr, Feb 13 2014

Status

approved