A111456 Pandigitals in some base (A061845) with an extra property: each number formed by the first i digits is divisible by i (digits in the pandigital base).

2, 108, 228, 13710, 44790, 6996920, 11128712, 12306056, 3816547290, 7838911147538198

Offset: 1

Author

Martin Fuller, Nov 15 2005

Keywords

• base
• nonn
• more
• hard
• changed

Comments

Finite? There are no more terms up to base 40. A probabilistic argument says higher bases are increasingly unlikely to produce a value.

There is no further term up to base=56; and no solution for base=60. Furthermore all bases are even: if the number formed by the first (base-1) digits is x, then x is divisible by (base-1) and x==base*(base-1)/2 mod (base-1), because the base-th digit is zero. From this the base is even. We can also see that if the i-th leftmost digit is d, then gcd(base,i)=gcd(base,d). To see this let g=gcd(base,i) and the number formed by the first i digit is x, then i divides x=k*base+d for some k, from this g divides d. And obviously g divides base, so g divides gcd(base,d), but it can't be larger than g, otherwise say gcd(base,d)=h>g, then in every h-th position we see a digit divisible by h, and the i-th digit is also divisible by h. This is a contradiction, there would be more than base/h digits divisible by h. - Robert Gerbicz, Mar 15 2016

Base corresponding to the terms: 2, 4, 4, 6, 6, 8, 8, 8, 10, 14. Terms written in its base: 10, 1230, 3210, 143250, 543210, 32541670, 52347610, 56743210, 3816547290, 9c3a5476b812d0 - Hans Havermann, May 26 2020

Subsequence of the terms of A256112 which are divisible by the base b in which they are pandigital (which is the least integer such that b^b > a(n)). In A256112 divisibility by i is required only for the numbers formed by the first i <= b-1 digits, while here it must also hold for i = b. - M. F. Hasler, May 26 2020

Examples

			E.g. 13710 = 143250[6] (i.e., in base 6) is pandigital and 14[6] = 10[10] is even, 143[6] = 63[10] is divisible by 3, 1432[6] = 380[10] is divisible by 4, etc.
3816547290 is a well-known example in base 10.
		

Crossrefs

Cf. A061845, A256112.

Programs

Python

def dgen(n, b):
    if n == 1:
        t = list(range(b))
        for i in range(1, b):
            u = list(t)
            u.remove(i)
            yield i, u
    else:
        for d, v in dgen(n-1, b):
            for g in v:
                k = d*b+g
                if not k % n:
                    u = list(v)
                    u.remove(g)
                    yield k, u
print([a for n in range(2, 15, 2) for a, b in dgen(n, n)]) # Chai Wah Wu, Jun 07 2015

Extensions

Keyword 'fini' is removed as finiteness is not proved. - Max Alekseyev, Dec 15 2014

Offset corrected to 1 by M. F. Hasler, May 27 2020