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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). 4
 2, 108, 228, 13710, 44790, 6996920, 11128712, 12306056, 3816547290, 7838911147538198 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 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 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 LINKS EXAMPLE E.g. 13710 = 143250 (i.e., in base 6) is pandigital and 14 = 10 is even, 143 = 63 is divisible by 3, 1432 = 380 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 print([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 Offset corrected to 1 by M. F. Hasler, May 27 2020 STATUS approved

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Last modified October 3 22:17 EDT 2022. Contains 357237 sequences. (Running on oeis4.)