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

1,2

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?

Links

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

Example

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.

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

Prog

(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

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.

Sequence in context: A051004 A032575 A038186 * A327453 A289791 A290386

Adjacent sequences: A118572 A118573 A118574 * A118576 A118577 A118578

Keyword

base,nonn

Author

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

Extensions

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

Status

approved