%I #16 Nov 26 2022 15:23:33

%S 18,48,110,111,234,306,342,396,486,576,756,792,1010,1100,1120,1164,

%T 1404,1548,1566,1740,1854,2106,2160,2376,2430,2502,2592,2640,2754,

%U 2790,2850,2880,3006,3060,3072,3078,3180,3330,3366,3420,3510,3564,3690

%N Numbers n that raised to the powers from 1 to k (with k>=1) are multiples of the sum of their digits (n raised to k+1 must not be a multiple). Case k=4.

%H Harvey P. Dale, <a href="/A135189/b135189.txt">Table of n, a(n) for n = 1..1000</a>

%F Positive integers n such that A195860(n) = 5.

%e 18^1 = 18 -> Sum_digits(18) = 9, and 18 is a multiple of 9.

%e 18^2 = 324 -> Sum_digits(324) = 9, and 324 is a multiple of 9.

%e 18^3 = 5832 -> Sum_digits(5832) = 18, and 5832 is a multiple of 18.

%e 18^4 = 104976 -> Sum_digits(104976) = 27, and 104976 is a multiple of 27

%e 18^5 = 1889568 -> Sum_digits(1889568) = 45, and 1889568 is not a multiple of 45.

%p readlib(log10); P:=proc(n,m) local a,i,k,w,x,ok; for i from 1 by 1 to n do a:=simplify(log10(i)); if not (trunc(a)=a) then ok:=1; x:=1; while ok=1 do w:=0; k:=i^x; while k>0 do w:=w+k-(trunc(k/10)*10); k:=trunc(k/10); od; if trunc(i^x/w)=i^x/w then x:=x+1; else if x-1=m then print(i); fi; ok:=0; fi; od; fi; od; end: P(2000,4);

%t msdQ[n_]:=Module[{t5=n^Range[5]},AllTrue[#/Total[IntegerDigits[#]]&/@ Most[ t5],IntegerQ]&&!Divisible[Last[t5],Total[IntegerDigits[Last[t5]]]]]; Select[ Range[ 4000],msdQ] (* _Harvey P. Dale_, Nov 26 2022 *)

%Y Cf. A135186 - A135202.

%K nonn,base

%O 1,1

%A _Paolo P. Lava_ & _Giorgio Balzarotti_, Nov 22 2007

%E More terms from _Max Alekseyev_, Sep 24 2011