A038186 Numbers divisible by the sum and product of their digits.

1, 2, 3, 4, 5, 6, 7, 8, 9, 12, 24, 36, 111, 112, 132, 135, 144, 216, 224, 312, 315, 432, 612, 624, 735, 1116, 1212, 1296, 1332, 1344, 1416, 2112, 2232, 2916, 3132, 3168, 3276, 3312, 4112, 4224, 6624, 6912, 8112, 9612, 11112, 11115, 11133, 11172, 11232

Offset: 1

Author

Felice Russo

Keywords

• nonn
• base
• nice
• look

Comments

The property "numbers divisible by the sum and product of their digits" leads to the Diophantine equation t*x1*x2*...*xr=s*(x1+x2+...+xr), where t and s are divisors of n; xi is from [1...9]. This corresponds to some arithmetic problems in geometry, see Sándor, 2002. - Ctibor O. Zizka, Mar 04 2008

Links

• David A. Corneth, Table of n, a(n) for n = 1..10352 (first 1000 terms from T. D. Noe)
• József Sándor, Geometric Theorems, Diophantine Equations and Arithmetic Functions, American Research Press, Rehoboth, 2002.

Crossrefs

Intersection of A005349 and A007602. - Reinhard Zumkeller, Apr 07 2011

Cf. A188641, A188642.

Programs

Haskell

import Data.List (elemIndices)
a038186 n = a038186_list !! (n-1)
a038186_list = map succ $ elemIndices 1
               $ zipWith (*) (map a188641 [1..]) (map a188642 [1..])
-- Reinhard Zumkeller, Apr 07 2011

Mathematica

dspQ[n_]:=Module[{idn=IntegerDigits[n],t},t=Times@@idn;t!=0 && Divisible[n,Total[idn]] && Divisible[n,t]]; Select[Range[11500],dspQ] (* Harvey P. Dale, Jul 11 2011 *)

PARI

for(n=1,10^4,d=digits(n);s=sumdigits(n);p=prod(i=1,#d,d[i]);if(p&&!(n%s+n%p),print1(n,", "))) \\ Derek Orr, Apr 29 2015

Python

from math import prod
def sd(n): return sum(map(int, str(n)))
def pd(n): return prod(map(int, str(n)))
def ok(n): return n%sd(n) == 0 and pd(n) and n%pd(n) == 0
def aupto(limit): return [m for m in range(1, limit+1) if ok(m)]
print(aupto(11233)) # Michael S. Branicky, Jan 28 2021

Formula

A188641(a(n)) * A188642(a(n)) = 1. - Reinhard Zumkeller, Apr 07 2011

Extensions

More terms from Patrick De Geest, Jun 15 1999