A064700 -id:A064700 - 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.

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

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

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?

			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.

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

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.

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

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

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

A118696 Semiprimes which are divisible by their multiplicative digital root.

Original entry on oeis.org

4, 6, 9, 15, 26, 34, 35, 62, 111, 115, 134, 278, 314, 355, 395, 398, 535, 694, 755, 1111, 1115, 1126, 1135, 1315, 1322, 1355, 1535, 1795, 2962, 3155, 3338, 3662, 3898, 3994, 4174, 4714, 5315, 6166, 6326, 6334, 6362, 6686, 6866, 6914, 6922, 7115, 7195, 7915

Offset: 1

Luc Stevens (lms022(AT)yahoo.com), May 20 2006

			134 is in the sequence because it is a semiprime and it is divisible by its multiplicative digital root, 2.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Intersection of A001358 and A064700.

spQ[n_] := Plus @@ Last /@ FactorInteger@n == 2; mdrQ[n_] := Mod[n, NestWhile[Times @@ IntegerDigits@# &, n, UnsameQ, All]] == 0; Select[ Range@9754, spQ@# && mdrQ@# &] (* Robert G. Wilson v, Aug 04 2006 *)
mdr[n_]:=Module[{c=NestWhile[Times@@IntegerDigits[#]&,n,#>9&]},If[c>0,c,Pi]]; Select[ Range[ 8000],PrimeOmega[#]==2&&Divisible[#,mdr[#]]&] (* Harvey P. Dale, Feb 27 2024 *)

A031347(n)= { local(resul,ncpy); if(n<10, return(n) ); ncpy=n; resul = ncpy % 10; ncpy = (ncpy - ncpy%10)/10; while( ncpy > 0, resul *= ncpy %10; ncpy = (ncpy - ncpy%10)/10; ); return(A031347(resul)); } { for(n=4,5000, if( bigomega(n)==2, dr=A031347(n); if(dr !=0 && n % dr == 0, print1(n,","); ); ); ); } \\ R. J. Mathar, May 23 2006

Corrected by R. J. Mathar, May 23 2006

More terms from Robert G. Wilson v, Aug 04 2006

A382402 Numbers divisible by the product of their digits (mod 10).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 12, 15, 24, 26, 34, 35, 37, 48, 55, 62, 64, 66, 72, 73, 75, 76, 78, 84, 88, 95, 96, 98, 99, 111, 112, 115, 126, 132, 134, 135, 136, 137, 144, 148, 155, 162, 164, 168, 172, 173, 175, 176, 184, 188, 192, 195, 196, 198, 199, 212, 216, 228, 232, 244, 248, 264, 266

Offset: 1

Enrique Navarrete, Mar 23 2025

Unlike A007602 and A064700, where there are no other primes besides 2, 3, 5, 7 and primes with repunits, this sequence contains other primes such as 37, 73 and 137.

The sequence has asymptotic density 0, since it contains no numbers with digit 5 and an even digit. - Robert Israel, Jun 01 2025

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A007602, A064700, A371281.

filter:= proc(n) local L,t;
  L:= convert(n,base,10);
  t:= convert(L,`*`) mod 10;
  t > 0 and n mod t = 0
end proc:
select(filter, [$1..1000]); # Robert Israel, Jun 01 2025

Select[Range[300], (prod = Mod[Times @@ IntegerDigits[#], 10]) > 0 && Divisible[#, prod] &] (* Amiram Eldar, Mar 23 2025 *)

isok(k) = my(p=lift(vecprod(apply(x->Mod(x, 10), digits(k))))); if (p, !(k % p)); \\ Michel Marcus, Mar 31 2025

from math import prod
def ok(n): return (p:=prod(map(int, str(n)))%10) > 0 and n%p == 0
print([k for k in range(300) if ok(k)]) # Michael S. Branicky, Mar 23 2025

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.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Python

Extensions

A118696 Semiprimes which are divisible by their multiplicative digital root.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

A382402 Numbers divisible by the product of their digits (mod 10).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Python