A118575 Dividuus numbers: numbers which are divisible by (1) the sum of their digits,(2) the product of their digits,(3) the digital root and (4) the multiplicative digital root.

1, 2, 3, 4, 5, 6, 7, 8, 9, 12, 24, 111, 112, 132, 135, 144, 216, 312, 315, 432, 612, 624, 1116, 1212, 1344, 1416, 2112, 2232, 3168, 3312, 4112, 4224, 6624, 8112, 11112, 11115, 11133, 11172, 11232, 11313, 11331, 11424, 11664, 12132, 12216, 12312, 12432

Offset: 1

Author

Luc Stevens (lms022(AT)yahoo.com), May 07 2006

Keywords

• base
• nonn

Comments

Dividuus : Latin for "divisible" Most of these numbers are even, but there are some odd numbers too. However, none of them seem to end on 7 (except for the obvious number 7 itself). Are there numbers in the sequence ending in 7?

Examples

			624 is in the sequence because (1) the sum of its digits is 6+4+2=12, (2) the product of its digits is 6*4*2=48, (3) the digital root is 3, (4) the multiplicative digital root is 6 and 624 is divisible by 12,48,3 and 6.
		

Links

• Chai Wah Wu, Table of n, a(n) for n = 1..2639

Crossrefs

Cf. A007953 (sum of digits), A007954 (product of digits), A010888 (digital root), A031347 (multiplicative digital root).

Intersection of A038186 and A064700 and A064807.

Subsequence of A005349, A007602, A038186, A064700, A064807.

Programs

Maple

filter:= proc(n)
local L, s,p;
  L:= convert(n,base,10);
  s:= convert(L,`+`);
  if n mod s <> 0 then return false fi;
  p:= convert(L,`*`);
  if p = 0 or n mod p <> 0 then return false fi;
  while s > 10 do
    s:= convert(convert(s,base,10),`+`);
  od:
  if n mod s <> 0 then return false fi;
  while p > 10 do
    p:= convert(convert(p, base, 10),`*`);
  od:
  p > 0 and n mod p = 0;
end proc:
select(filter, [$1..10^4]); # Robert Israel, Aug 24 2014

Python

from operator import mul
from functools import reduce
from gmpy2 import t_mod, mpz
def A031347(n):
    while n > 9:
        n = reduce(mul, (int(d) for d in str(n)))
    return n
A118575 = [n for n in range(1, 10**9) if A031347(n) and not
           (str(n).count('0') or t_mod(n, (1+t_mod((n-1), 9))) or
           t_mod(n, A031347(n)) or t_mod(n,sum((mpz(d) for d in str(n))))
           or t_mod(n, reduce(mul,(mpz(d) for d in str(n)))))]
# Chai Wah Wu, Aug 26 2014

Extensions

Inserted a(17)=216 by Chai Wah Wu, Aug 24 2014