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

Author

M. F. Hasler, Jan 03 2014

Keywords

• nonn
• base
• more

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.

Examples

			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.
		

Programs

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(Vec(d)*vector(n,i,10^(n-i))~", ")))

Formula

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