A118575 - OEIS

%I #22 May 28 2020 17:28:07

%S 1,2,3,4,5,6,7,8,9,12,24,111,112,132,135,144,216,312,315,432,612,624,

%T 1116,1212,1344,1416,2112,2232,3168,3312,4112,4224,6624,8112,11112,

%U 11115,11133,11172,11232,11313,11331,11424,11664,12132,12216,12312,12432

%N 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.

%C 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?

%H Chai Wah Wu, <a href="/A118575/b118575.txt">Table of n, a(n) for n = 1..2639</a>

%e 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.

%p filter:= proc(n)

%p local L, s,p;

%p   L:= convert(n,base,10);

%p   s:= convert(L,`+`);

%p   if n mod s <> 0 then return false fi;

%p   p:= convert(L,`*`);

%p   if p = 0 or n mod p <> 0 then return false fi;

%p   while s > 10 do

%p     s:= convert(convert(s,base,10),`+`);

%p   od:

%p   if n mod s <> 0 then return false fi;

%p   while p > 10 do

%p     p:= convert(convert(p, base, 10),`*`);

%p   od:

%p   p > 0 and n mod p = 0;

%p end proc:

%p select(filter, [$1..10^4]); # _Robert Israel_, Aug 24 2014

%o (Python)

%o from operator import mul

%o from functools import reduce

%o from gmpy2 import t_mod, mpz

%o def A031347(n):

%o     while n > 9:

%o         n = reduce(mul, (int(d) for d in str(n)))

%o     return n

%o A118575 = [n for n in range(1, 10**9) if A031347(n) and not

%o            (str(n).count('0') or t_mod(n, (1+t_mod((n-1), 9))) or

%o            t_mod(n, A031347(n)) or t_mod(n,sum((mpz(d) for d in str(n))))

%o            or t_mod(n, reduce(mul,(mpz(d) for d in str(n)))))]

%o # _Chai Wah Wu_, Aug 26 2014

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

%Y Intersection of A038186 and A064700 and A064807.

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

%K base,nonn

%O 1,2

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

%E Inserted a(17)=216 by _Chai Wah Wu_, Aug 24 2014