cp's OEIS Frontend

A108827 Numbers k that divide the sum of the digits of k^k.

%I A108827 #26 Feb 24 2024 01:11:45
%S A108827 1,2,3,9,18,27,54,90,108,163,197,254,432,1292,2202,9648,10347,16596,
%T A108827 17203,46188,46992,77121,130082,167410,216546,596277
%N A108827 Numbers k that divide the sum of the digits of k^k.
%C A108827 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
%C A108827 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 A108827 3^3 = 27; 2 + 7 = 9; 9 mod 3 = 0.
%e A108827 9^9 = 387420489; 3 + 8 + 7 + 4 + 2 + 4 + 8 + 9 = 45; 45 mod 9 = 0.
%e A108827 432 is a term because the sum of the digits of 432^432 = 5184 is divisible by 432.
%p A108827 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_
%t A108827 Do[If[Mod[Plus @@ IntegerDigits[n^n], n] == 0, Print[n]], {n, 1, 10000}]
%t A108827 Select[Range[600000],Divisible[Total[IntegerDigits[#^#]],#]&] (* _Harvey P. Dale_, Jan 28 2017 *)
%Y A108827 Cf. A125526, A125724, A108825.
%K A108827 more,nonn,base
%O A108827 1,2
%A A108827 _Ryan Propper_, Jul 11 2005
%E A108827 a(16)-a(19) from Simon Nickerson (simonn(AT)maths.bham.ac.uk) and _Emeric Deutsch_, Jul 15 2005
%E A108827 a(20)-a(22) from _Ray Chandler_, Jul 25 2005
%E A108827 Edited by _N. J. A. Sloane_, Apr 27 2008 at the suggestion of _Stefan Steinerberger_
%E A108827 a(23) from _Robert G. Wilson v_, May 17 2008
%E A108827 a(24) from _Robert G. Wilson v_, May 19 2008
%E A108827 a(25)-a(26) from _Lars Blomberg_, Jul 09 2011