cp's OEIS Frontend

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.

%I A118575 #22 May 28 2020 17:28:07
%S A118575 1,2,3,4,5,6,7,8,9,12,24,111,112,132,135,144,216,312,315,432,612,624,
%T A118575 1116,1212,1344,1416,2112,2232,3168,3312,4112,4224,6624,8112,11112,
%U A118575 11115,11133,11172,11232,11313,11331,11424,11664,12132,12216,12312,12432
%N 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.
%C A118575 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 A118575 Chai Wah Wu, <a href="/A118575/b118575.txt">Table of n, a(n) for n = 1..2639</a>
%e A118575 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 A118575 filter:= proc(n)
%p A118575 local L, s,p;
%p A118575   L:= convert(n,base,10);
%p A118575   s:= convert(L,`+`);
%p A118575   if n mod s <> 0 then return false fi;
%p A118575   p:= convert(L,`*`);
%p A118575   if p = 0 or n mod p <> 0 then return false fi;
%p A118575   while s > 10 do
%p A118575     s:= convert(convert(s,base,10),`+`);
%p A118575   od:
%p A118575   if n mod s <> 0 then return false fi;
%p A118575   while p > 10 do
%p A118575     p:= convert(convert(p, base, 10),`*`);
%p A118575   od:
%p A118575   p > 0 and n mod p = 0;
%p A118575 end proc:
%p A118575 select(filter, [$1..10^4]); # _Robert Israel_, Aug 24 2014
%o A118575 (Python)
%o A118575 from operator import mul
%o A118575 from functools import reduce
%o A118575 from gmpy2 import t_mod, mpz
%o A118575 def A031347(n):
%o A118575     while n > 9:
%o A118575         n = reduce(mul, (int(d) for d in str(n)))
%o A118575     return n
%o A118575 A118575 = [n for n in range(1, 10**9) if A031347(n) and not
%o A118575            (str(n).count('0') or t_mod(n, (1+t_mod((n-1), 9))) or
%o A118575            t_mod(n, A031347(n)) or t_mod(n,sum((mpz(d) for d in str(n))))
%o A118575            or t_mod(n, reduce(mul,(mpz(d) for d in str(n)))))]
%o A118575 # _Chai Wah Wu_, Aug 26 2014
%Y A118575 Cf. A007953 (sum of digits), A007954 (product of digits), A010888 (digital root), A031347 (multiplicative digital root).
%Y A118575 Intersection of A038186 and A064700 and A064807.
%Y A118575 Subsequence of A005349, A007602, A038186, A064700, A064807.
%K A118575 base,nonn
%O A118575 1,2
%A A118575 Luc Stevens (lms022(AT)yahoo.com), May 07 2006
%E A118575 Inserted a(17)=216 by _Chai Wah Wu_, Aug 24 2014