A235133 Numbers whose (decimal) digits are a permutation of 1...n for some n, such that for all k in {1,...,n} the first k digits form a number divisible by k, when considered as representation in base n+1.

1, 123, 321, 14325, 54321, 3254167, 5234761, 5674321, 381654729

Offset

1,2

Comments

See sequence A163574 for further discussion, in particular the proof that n must be odd, and that there is no further term with less than 13 digits. There are several options for representing the terms with 13 and more digits, e.g., continuing to write the number d[1]*10^(n-1)+...+d[n]*10^0, or concatenation of the "digits", written in base 10, even when they are larger than 9.

See sequence A235164 for the value of the terms when the n digits are read in base n+1, as it is also done in A163574.

Links

Table of n, a(n) for n=1..9.

Formula

a(n) = concatenation of digits of x = A235164(n) when written in base b = A055642(x)+1 = (# digits of x)+1.

Example

a(2) = 123 is in the sequence, because in base 4, 12[4]=6 is divisible by 2 and 123[4] = 27 = A235164(2) is divisible by 3. The same is the case for 321, where 32[4]=14 is even and 321[4] = 57 = A235164(3) is divisible by 3.
For the 9-digit term 381654729, the initial digits are to be interpreted in base 10: 38, 318, ..., 381654729 are divisible by 2, 3, ..., 9, respectively.

Prog

(PARI) for(n=1, 9, p=vector(n, i, (n+1)^(i-1))); for(k=0, n!-1, d=numtoperm(n, k); for(j=2, n, sum(i=1, j, d[i]*p[j-i+1])%j &&next(2)); print1(d*vector(n, i, 10^(n-i))~", ")))

Crossrefs

Sequence in context: A216382 A037139 A269751 * A179612 A091331 A248556

Adjacent sequences: A235130 A235131 A235132 * A235134 A235135 A235136

Keyword

nonn,base

Author

M. F. Hasler, Jan 03 2014

Status

approved