

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




OFFSET

0,1


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 (base1) digits is x, then x is divisible by (base1) and x==base*(base1)/2 mod (base1), because the baseth digit is zero. From this the base is even. We can also see that if the ith 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 hth position we see a digit divisible by h, and the ith 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


LINKS

Table of n, a(n) for n=0..9.


EXAMPLE

E.g. 13710=143250 (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 wellknown example in base 10.


PROG

(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(n1, b):
............for g in v:
................k = d*b+g
................if not k % n:
....................u = list(v)
....................u.remove(g)
....................yield k, u
A111456_list = [a for n in range(2, 15, 2) for a, b in dgen(n, n)] # Chai Wah Wu, Jun 07 2015


CROSSREFS

Cf. A061845, A256112.
Sequence in context: A267550 A224819 A156502 * A157067 A287153 A247098
Adjacent sequences: A111453 A111454 A111455 * A111457 A111458 A111459


KEYWORD

base,nonn,more,hard


AUTHOR

Martin Fuller, Nov 15 2005


EXTENSIONS

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


STATUS

approved



