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).

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, 216, 243, 324, 378, 405, 432, 486, 648, 756, 864, 972, 1296, 1458, 1944, 2916, 3402, 4374, 5832, 6804, 7290, 8748, 11664, 13122, 13608, 15552, 17496, 23328, 26244

Offset

1,2

Comments

The sequence is finite with a(15095), a 1434-digit number, being the final term. - Hans Havermann and Ray Chandler, Jan 21 2014

Links

Donovan Johnson, Table of n, a(n) for n = 1..235 (terms < 10^17)

Hans Havermann and Ray Chandler, Table of n, a(n) for n = 1..15095 (9.3 MB file)

Kornel, Ojciec i Syn (Polish) "Father and Son", mentions the term 216.

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, As much as I hate "base" sequences..., Copies of various posts to the Sequence Fans Mailing List, Circa January 2014. Assembled by N. J. A. Sloane, Dec 23 2024

Example

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.

Mathematica

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 *)

Prog

(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 */

Crossrefs

Cf. A005349, A097569, A235600, A235601, A236295, A236362, A236363, A236385.

Sequence in context: A193455 A356349 A343680 * A334416 A217973 A097518

Adjacent sequences: A114437 A114438 A114439 * A114441 A114442 A114443

Keyword

nonn,base,fini,full

Author

Piotr K. Olszewski (piotrkornelolszewski(AT)poczta.onet.pl), Feb 14 2006

Extensions

Offset corrected by Donovan Johnson, Apr 09 2013

a(54)-a(235) from Donovan Johnson, Apr 09 2013

a(236)-a(15095) from Hans Havermann and Ray Chandler, Jan 21 2014

Status

approved