A226171 Smallest base in which n is not Niven (or zero if n is Niven in every base).

0, 0, 2, 0, 2, 0, 2, 6, 2, 4, 2, 8, 2, 2, 2, 6, 2, 8, 2, 7, 5, 2, 2, 14, 2, 2, 2, 2, 2, 2, 2, 6, 2, 3, 2, 8, 2, 2, 2, 12, 2, 3, 2, 2, 2, 2, 2, 14, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 8, 2, 2, 2, 6, 2, 3, 2, 3, 3, 2, 2, 14, 2, 2, 2, 2, 2, 2, 2, 8, 5, 2, 2, 5, 2, 2

Offset

1,3

Comments

Niven numbers (in base b) are divisible by the sum of their digits (in base b).

Questions: are 1, 2, 4 and 6 the only zeros in this sequence? Where are the records or high water marks?

From Bert Dobbelaere, Oct 08 2018: (Start)

1,2,4,6 are the only numbers that are Niven in every base.

Proof: Suppose n is Niven in every base, then consider the base-b representations of n for (n/2) < b <= n. These are all 2-digit numbers with 1 as 1st digit and (n-b) as last digit. Then 1+n-b is a divisor of n for all b, meaning that all numbers between 1 up to n/2 are divisors of n. Clearly there are no such numbers larger than 6.

a(n) < 60 for n < 10^13.

(End)

Links

T. D. Noe, Table of n, a(n) for n = 1..10000

Example

The sum of digits of 24 in bases 1 through 14 are:  24, 2, 4, 3, 8, 4, 6, 3, 8, 6, 4, 2, 12, 11.  24 is divisible by all these numbers except the last one; therefore a(24) = 14.

Mathematica

Table[b = 2; While[s = Total[IntegerDigits[n, b]]; s < n && Mod[n, s] == 0, b++]; If[s == n, b = 0]; b, {n, 100}] (* T. D. Noe, May 30 2013 *)

Prog

(PARI) a(n) = {for (b=2, n-1, if (frac(n/sumdigits(n, b)), return(b)); ); 0; } \\ Michel Marcus, Oct 23 2018

Crossrefs

Cf. A049445, A118363, A005349, A144261, A144262, A226169.

Cf. A225427 (least Niven number for all bases from 1 to n).

Sequence in context: A281981 A159006 A291968 * A284280 A278520 A239246

Adjacent sequences: A226168 A226169 A226170 * A226172 A226173 A226174

Keyword

nonn,base

Author

Sergio Pimentel, May 29 2013

Status

approved