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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).

%I A111456 #38 Sep 04 2025 08:40:55
%S A111456 2,108,228,13710,44790,6996920,11128712,12306056,3816547290,
%T A111456 7838911147538198
%N 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).
%C A111456 Finite? There are no more terms up to base 40. A probabilistic argument says higher bases are increasingly unlikely to produce a value.
%C A111456 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
%C A111456 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
%C A111456 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
%e A111456 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.
%e A111456 3816547290 is a well-known example in base 10.
%o A111456 (Python)
%o A111456 def dgen(n, b):
%o A111456     if n == 1:
%o A111456         t = list(range(b))
%o A111456         for i in range(1, b):
%o A111456             u = list(t)
%o A111456             u.remove(i)
%o A111456             yield i, u
%o A111456     else:
%o A111456         for d, v in dgen(n-1, b):
%o A111456             for g in v:
%o A111456                 k = d*b+g
%o A111456                 if not k % n:
%o A111456                     u = list(v)
%o A111456                     u.remove(g)
%o A111456                     yield k, u
%o A111456 print([a for n in range(2, 15, 2) for a, b in dgen(n, n)]) # _Chai Wah Wu_, Jun 07 2015
%Y A111456 Cf. A061845, A256112.
%K A111456 base,nonn,more,hard,changed
%O A111456 1,1
%A A111456 _Martin Fuller_, Nov 15 2005
%E A111456 Keyword 'fini' is removed as finiteness is not proved. - _Max Alekseyev_, Dec 15 2014
%E A111456 Offset corrected to 1 by _M. F. Hasler_, May 27 2020