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

1,2

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.

Formula

A188641(a(n)) * A188642(a(n)) = 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 *)

Prog

(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
(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

Crossrefs

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

Cf. A188641, A188642.

Sequence in context: A001102 A051004 A032575 * A118575 A327453 A289791

Adjacent sequences: A038183 A038184 A038185 * A038187 A038188 A038189

Keyword

nonn,base,nice,look

Author

Felice Russo

Extensions

More terms from Patrick De Geest, Jun 15 1999

Status

approved