A163574 Decimal expansion of smallest zeroless pandigital number in base n such that each k-digit substring (1 <= k <= n-1 = number of base-n digits) starting from the left, is divisible by k (or 0 if none exists).

1, 0, 27, 0, 2285, 0, 874615, 0, 381654729, 0, 0, 0, 559922224824157, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 2

Author

Gaurav Kumar, Jul 31 2009

Keywords

• nonn
• base
• more

Comments

Sequence gives smallest term with desired property.

For n=2 and 10, there is only one such number.

For n=4, there are 2 solutions: 27 and 57, the latter 321(4).

For n=6, there are 2 solutions: 2285 and 7465, the latter 54321(6).

For n=8, there are 3 solutions: 874615, 1391089 and 1538257, these last two being 5234761(8) and 5674321(8).

There are no solutions for a number system of base n, if n is odd. For a solution the sum of the digits is always (n-1)*n/2. A solution is always divisible by n-1. This is only possible if the sum of the digits is divisible by n-1. As a consequence, n/2 has to be an integer and therefore n has to be even (translated from 2nd link from German by web page author, Werner Brefeld). - Michel Marcus, Dec 09 2013

Is it true that a(n) = 0 for n > 14? - Chai Wah Wu, Jun 07 2015

For a solution in base n, if the k-th digit from the left is d, then gcd(d,n) = gcd(k,n). In particular, digits in even positions are even, and digits in odd positions are odd. - David Radcliffe, Apr 24 2016

If a(n) <> 0 and is expressed in base n, the middle digit must be n/2. - Thomas Kaeding, Sep 01 2019

Examples

			a(3) = 0, since the 2 possible zeroless numbers, 12 and 21 in base 3, are both odd numbers, so do not satisfy the condition for k=2.
a(4) = 27, that is 123 in base 4, such that 1, 12, and 123 are respectively divisible by 1, 2 and 3.
Expansion of each term in the corresponding base : 27 = 123 (4); 2285 = 14325 (6); 874615 = 3254167 (8); 381654729 = 381654729 (10); 559922224824157 = 9C3A5476B812D (14).
		

Links

• Blaine, How about a math puzzle?
• Werner Brefeld, Neunstellige Zahl und Teilbarkeit (in German).
• Albert Franck, Puzzles, see item 7.

Programs

PARI

a(n) = {n--; for (j=0, n!-1, perm = numtoperm(n, j); ok = 1; for (i=1, n, v = sum(k=1, i, perm[k]*(n+1)^(i-k)); if ((v % i), ok=0; break;);); if (ok, return(v)););} \\ Michel Marcus, Dec 01 2013

PARI

chka(n, b) = {digs = digits(n, b); for (i=1, #digs, v = sum(k=1, i, digs[k]*b^(i-k)); print(v, ": ", v/i); if (v % i, return (0));); return (1);} \\ Michel Marcus, Dec 02 2013

PARI

okdigits(v, i) = {for (j=1, i-1, if (v[i] == v[j], return (0));); return (1);}
a(n) = {b = n; n--; v = vector(n, i, 0); i = 1; while (1, v[i]++; while (v[i] > n, v[i] = 0; i --; if (i==0, return (0)); v[i]++); curv = sum (j=1, i, v[j]*(b^(i-j))); if (! (curv % i), if (okdigits(v, i), if (i == n, return (sum (j=1, n, v[j]*(b^(n-j))))); i++;);););} \\ Michel Marcus, Dec 08 2013

Python

def vgen(n, b):
    if n == 1:
        t = list(range(1, b))
        for i in range(1, b):
            u = list(t)
            u.remove(i)
            yield i, u
    else:
        for d, v in vgen(n-1, b):
            for g in v:
                k = d*b+g
                if not k % n:
                    u = list(v)
                    u.remove(g)
                    yield k, u
def A163574(n):
    for a, b in vgen(n-1, n):
        return a
    return 0 # Chai Wah Wu, Jun 07 2015

Formula

a(2n+1) = 0 (see proof in comment). - Michel Marcus, Dec 09 2013

Extensions

Corrected and edited by Michel Marcus, Dec 02 2013

More terms from Michel Marcus, Dec 09 2013

a(31)-a(41) from Chai Wah Wu, Jun 07 2015

a(42)-a(49) from David Radcliffe, Apr 24 2016

a(50)-a(53) from Kevin Thomas, Jun 11 2019

a(54)-a(57) from Thomas Kaeding, Sep 03 2019