cp's OEIS Frontend

A114440 Numbers which divided by the sum of their digits (Harshad or Niven numbers) give integers which are also divisible by the sum of their digits (until a single-digit Harshad remains).

%I A114440 #33 Dec 23 2024 23:22:17
%S A114440 1,2,3,4,5,6,7,8,9,12,18,21,24,27,36,42,45,48,54,63,72,81,84,108,162,
%T A114440 216,243,324,378,405,432,486,648,756,864,972,1296,1458,1944,2916,3402,
%U A114440 4374,5832,6804,7290,8748,11664,13122,13608,15552,17496,23328,26244
%N A114440 Numbers which divided by the sum of their digits (Harshad or Niven numbers) give integers which are also divisible by the sum of their digits (until a single-digit Harshad remains).
%C A114440 The sequence is finite with a(15095), a 1434-digit number, being the final term. - _Hans Havermann_ and _Ray Chandler_, Jan 21 2014
%H A114440 Donovan Johnson, <a href="/A114440/b114440.txt">Table of n, a(n) for n = 1..235</a> (terms < 10^17)
%H A114440 Hans Havermann and Ray Chandler, <a href="http://chesswanks.com/seq/b114440.txt">Table of n, a(n) for n = 1..15095</a> (9.3 MB file)
%H A114440 Kornel, <a href="http://forum.gazeta.pl/forum/72,2.html?f=514&amp;w=36333926&amp;a=36333926">Ojciec i Syn</a> (Polish) "Father and Son", mentions the term 216.
%H A114440 David W. Wilson, Ray Chandler, Alonso Del Arte, M. F. Hasler, Hans Havermann, Alex Meiburg, N. J. A. Sloane, Hugo Van Der Sanden, and Allan Wechsler, <a href="/A235601/a235601.txt">As much as I hate "base" sequences...</a>, Copies of various posts to the Sequence Fans Mailing List, Circa January 2014. Assembled by _N. J. A. Sloane_, Dec 23 2024
%e A114440 The number 216 is a term of the sequence because it is divisible by the sum of its digits: 2+1+6=9; 216/9=24. Also, the successive quotients are divisible by the sum of their digits, until a single-digit Harshad remains: 24: 2+4=6; 24/6=4 and 4: 4/4=1.
%t A114440 s=w={1}; Do[t={}; Do[v=s[[k]]; u={}; Do[If[Total[IntegerDigits[c*v]]==c, AppendTo[u,c*v]], {c,2,7000}]; t=Join[t,u], {k,Length[s]}]; s=Sort[t]; w=Join[w,s], {440}]; Union[w] (* _Hans Havermann_, Jan 21 2014 *)
%o A114440 (PARI) v=vector(118); for(n=1, 9, v[n]=n; print1(n ", ")); c=9; for(n=10, 10^9, d=length(Str(n)); m=n; s=0; for(j=1, d, s=s+m%10; m=m\10); if(s==1, next); if(n%s==0, m=n/s, next); forstep(j=c, 1, -1, if(v[j]<=m, if(v[j]==m, c++; v[c]=n; print1(n ", ")); next(2)))) /* _Donovan Johnson_, Apr 09 2013 */
%Y A114440 Cf. A005349, A097569, A235600, A235601, A236295, A236362, A236363, A236385.
%K A114440 nonn,base,fini,full
%O A114440 1,2
%A A114440 Piotr K. Olszewski (piotrkornelolszewski(AT)poczta.onet.pl), Feb 14 2006
%E A114440 Offset corrected by _Donovan Johnson_, Apr 09 2013
%E A114440 a(54)-a(235) from _Donovan Johnson_, Apr 09 2013
%E A114440 a(236)-a(15095) from _Hans Havermann_ and _Ray Chandler_, Jan 21 2014