OFFSET
1,2
COMMENTS
Especially for larger terms k not divisible by 10, we can expect 4.5 times the number of digits in k^k to be close to some integer multiple (m) of k, so k should occur near 100^(m/9). E.g., for m = 10, 11, ..., 16, approximate (and corresponding actual) values would be 167 (163, 197), 278 (254), 464 (432), 774 (none), 1292 (1292), 2154 (2022) and 3594 (none). Larger terms k ending with exactly j zeros would be expected to occur near k = 10^j * 100^(m/9) for some integer m. - Jon E. Schoenfield, Jun 09 2007
The quotients are 1, 2, 3, 5, 6, 7, 7, 4, 9, 10, 10, 11, 12, 14, 15, 18, 18, 19, 19, 21, 21, 22, 23, 19, 24, 26.
EXAMPLE
3^3 = 27; 2 + 7 = 9; 9 mod 3 = 0.
9^9 = 387420489; 3 + 8 + 7 + 4 + 2 + 4 + 8 + 9 = 45; 45 mod 9 = 0.
432 is a term because the sum of the digits of 432^432 = 5184 is divisible by 432.
MAPLE
a:=proc(n) local nn: nn:=convert(n^n, base, 10): if type(add(nn[j], j=1..nops(nn))/n, integer)=true then n else fi end: seq(a(n), n=1..2000); # Emeric Deutsch
MATHEMATICA
Do[If[Mod[Plus @@ IntegerDigits[n^n], n] == 0, Print[n]], {n, 1, 10000}]
Select[Range[600000], Divisible[Total[IntegerDigits[#^#]], #]&] (* Harvey P. Dale, Jan 28 2017 *)
CROSSREFS
KEYWORD
more,nonn,base
AUTHOR
Ryan Propper, Jul 11 2005
EXTENSIONS
a(16)-a(19) from Simon Nickerson (simonn(AT)maths.bham.ac.uk) and Emeric Deutsch, Jul 15 2005
a(20)-a(22) from Ray Chandler, Jul 25 2005
Edited by N. J. A. Sloane, Apr 27 2008 at the suggestion of Stefan Steinerberger
a(23) from Robert G. Wilson v, May 17 2008
a(24) from Robert G. Wilson v, May 19 2008
a(25)-a(26) from Lars Blomberg, Jul 09 2011
STATUS
approved