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A108827 Numbers n such that n divides the sum of the digits of n^n. 3

%I

%S 1,2,3,9,18,27,54,90,108,163,197,254,432,1292,2202,9648,10347,16596,

%T 17203,46188,46992,77121,130082,167410,216546,596277

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

%C 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

%C 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.

%e 3^3=27; 2+7=9; (9 mod 3)=0

%e 9^9=387420489; 3+8+7+4+2+4+8+9=45; (45 mod 9)=0

%e 432 is a term because the sum of the digits of 432^432 = 5184 is divisible by 432.

%p 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 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

%t Do[If[Mod[Plus @@ IntegerDigits[n^n], n] == 0, Print[n]], {n, 1, 10000}]

%t Select[Range[600000],Divisible[Total[IntegerDigits[#^#]],#]&] (* _Harvey P. Dale_, Jan 28 2017 *)

%Y Cf. A125526, A125724, A108825.

%K more,nonn,base

%O 1,2

%A _Ryan Propper_, Jul 11 2005

%E a(16)-a(19) from Simon Nickerson (simonn(AT)maths.bham.ac.uk) and _Emeric Deutsch_, Jul 15 2005

%E a(20)-a(22) from _Ray Chandler_, Jul 25 2005

%E Edited by _N. J. A. Sloane_, Apr 27 2008 at the suggestion of _Stefan Steinerberger_

%E a(23) from _Robert G. Wilson v_, May 17 2008

%E a(24) from _Robert G. Wilson v_, May 19 2008

%E a(25)-a(26) from _Lars Blomberg_, Jul 09 2011

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Last modified September 29 21:59 EDT 2020. Contains 337432 sequences. (Running on oeis4.)