A068187 a(n) is the smallest number such that the product of its decimal digits equals n^n, or 0 if no solutions exist.

1, 4, 39, 488, 55555, 88999, 7777777, 88888888, 999999999, 25555555555888, 0, 88888888999999, 0, 4777777777777778888, 35555555555555559999999, 2888888888888888888888, 0, 888888999999999999999999, 0, 2555555555555555555558888888888888, 37777777777777777777779999999999

Offset: 1

Author

Labos Elemer, Feb 18 2002

Keywords

• base
• nonn

Comments

a(n) = 0 if and only if n has a prime factor > 7. If n > 1 has no prime factor > 7, let n^n = 2^a*3^b*5^c*7^d. Let m(x) denote the number of digit x in a(n). Then a(n) is a number whose digits are nondecreasing and defined as follows. m(2) = 1 if a mod 3 == 1 and 0 otherwise, m(3) = 1 if b mod 2 == 1 and 0 otherwise, m(4) = 1 if a mod 3 == 2 and 0 otherwise, m(5) = c, m(6) = 0, m(7) = d, m(8) = floor(a/3), m(9) = floor(b/2). - Chai Wah Wu, Aug 12 2017

Links

• Chai Wah Wu, Table of n, a(n) for n = 1..200

Crossrefs

Cf. A000312, A067734, A068183, A068184, A068185, A068186, A068188, A068190.

Programs

Python

from sympy import factorint
def A068187(n):
    if n == 1:
        return 1
    pf = factorint(n)
    return 0 if max(pf) > 7 else int(''.join(sorted(''.join(str(a)*(n*b) for a,b in pf.items()).replace('222','8').replace('22','4').replace('33','9')))) # Chai Wah Wu, Aug 13 2017

Extensions

Edited by Dean Hickerson and Henry Bottomley, Mar 07 2002