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A111456
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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).
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4
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OFFSET
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1,1
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COMMENTS
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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
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LINKS
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EXAMPLE
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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.
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PROG
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(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
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CROSSREFS
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KEYWORD
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base,nonn,more,hard
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AUTHOR
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EXTENSIONS
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Keyword 'fini' is removed as finiteness is not proved. - Max Alekseyev, Dec 15 2014
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STATUS
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approved
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