login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A109675 Numbers k such that the sum of the digits of (k^k - 1) is divisible by k. 0

%I #17 Jun 27 2023 07:07:38

%S 1,4,5,10,25,50,100,446,1000,9775,10000,100000,995138,996544,998866,

%T 1000000

%N Numbers k such that the sum of the digits of (k^k - 1) is divisible by k.

%C k = 10^m is a term of the sequence for all m >= 0. Proof: Let k = 10^m for some nonnegative integer m. Then k^k - 1 has m*10^m 9's and no other digits, so its digits sum to 9*m*10^m = 9*m*k, a multiple of k.

%e The digits of 9775^9775 - 1 sum to 175950 and 175950 is divisible by 9775, so 9775 is in the sequence.

%p sumdigs:= n -> convert(convert(n,base,10),`+`);

%p select(n -> sumdigs(n^n-1) mod n = 0, [$1..10^5]); # _Robert Israel_, Dec 03 2014

%t Do[k = n^n - 1; s = Plus @@ IntegerDigits[k]; If[Mod[s, n] == 0, Print[n]], {n, 1, 10^5}]

%o (Python)

%o A109675_list = [n for n in range(1,10**4) if not sum([int(d) for d in str(n**n-1)]) % n]

%o # _Chai Wah Wu_, Dec 03 2014

%Y Cf. A007953, A048861.

%K base,hard,more,nonn

%O 1,2

%A _Ryan Propper_, Aug 06 2005

%E a(13)-a(16) from _Michael S. Branicky_, Jun 25 2023

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified March 28 09:04 EDT 2024. Contains 371240 sequences. (Running on oeis4.)