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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
2, 108, 228, 13710, 44790, 6996920, 11128712, 12306056, 3816547290, 7838911147538198 (list; graph; refs; listen; history; text; internal format)
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 (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

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 well-known 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(n-1, 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 A247098 A245056

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

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Last modified March 23 21:31 EDT 2017. Contains 283985 sequences.