

A108827


Numbers n such that n divides the sum of the digits of n^n.


3



1, 2, 3, 9, 18, 27, 54, 90, 108, 163, 197, 254, 432, 1292, 2202, 9648, 10347, 16596, 17203, 46188, 46992, 77121, 130082, 167410, 216546, 596277
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OFFSET

1,2


COMMENTS

Especially for larger terms n not divisible by 10, we can expect 4.5 times the number of digits in n^n to be close to some integer multiple (m) of n, so n 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 n ending with exactly j zeros would be expected to occur near n = 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.


LINKS

Table of n, a(n) for n=1..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
P:=proc(n) local i, k, w; for i from 1 by 1 to n do w:=0; k:=i^i; while k>0 do w:=w+k(trunc(k/10)*10); k:=trunc(k/10); od; if (w mod i)=0 then print(i); fi; od; end: P(1000); # Paolo P. Lava and Giorgio Balzarotti, Jun 04 2007


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

Cf. A125526, A125724, A108825.
Sequence in context: A234646 A065965 A296052 * A298347 A113201 A089753
Adjacent sequences: A108824 A108825 A108826 * A108828 A108829 A108830


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



