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%I #10 Sep 08 2022 08:45:30
%S 1,2,3,4,5,6,7,8,9,11,12,14,111,112,128,216,512,1111,1116,1127,1211,
%T 1274,5187,8151,11111,15125,41111,111111,111188,111511,141151,1111111,
%U 1111121,1112111,1211111,11111111,11111175,11211291,71111117,111111111
%N Numbers n that are divisible by d, where a_i for 1 <= i <= k are the digits of n and d = Product_{i=1..k}Sum_of_digits_of_(a_i^k).
%C Sequence is infinite since it contains all repunits > 0; are there infinitely many other terms?
%e 5, 1, 8, 7 are the four digits of 5187. 5^4 = 625 and 6+2+5 = 13; 1^4 = 1; 8^4 = 4096 and 4+0+9+6 = 19; 7^4 = 2401 and 2+4+0+1 = 7. Since 13*1*19*7 = 1729 divides 5187 = 1729*3, 5187 is in the sequence.
%p P:=proc(n) local i,j,k,w,y,prod,cont; for i from 1 by 1 to n do w:=0; k:=i; cont:=0; while k>0 do k:=trunc(k/10); cont:=cont+1; od; k:=i; prod:=1; for j from 1 to cont do w:=(k-(trunc(k/10)*10))^cont; y:=0; while w>0 do y:=y+w-(trunc(w/10)*10); w:=trunc(w/10); od; prod:=prod*y; k:=trunc(k/10); od; if prod>0 then if trunc(i/prod)=i/prod then print(i); fi; fi; od; end: P(200000);
%t nddQ[n_]:=Module[{c=Times@@(Total[IntegerDigits[#]]&/@(IntegerDigits[ n]^ IntegerLength[n]))},c!=0&&Divisible[n,c]]; Select[Range[111111111], nddQ] (* _Harvey P. Dale_, Jun 05 2016 *)
%o (Magma) [ n: n in [1..112000000] | p gt 0 and n mod p eq 0 where p is &*[ &+Intseq(x^#k, 10): x in k ] where k is Intseq(n, 10) ]; // _Klaus Brockhaus_, Apr 25 2007
%Y Cf. A002275 (repunits), A005188 (Armstrong numbers).
%K nonn,base
%O 1,2
%A _Paolo P. Lava_ and _Giorgio Balzarotti_, Apr 18 2007
%E Edited and a(36) to a(40) added by _Klaus Brockhaus_, Apr 25 2007
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