A247145 Composite numbers such that the product of the number's proper divisors is divisible by the sum of the number's proper divisors.

6, 12, 24, 28, 40, 42, 56, 60, 90, 120, 140, 153, 216, 234, 270, 290, 360, 440, 496, 522, 568, 585, 588, 672, 708, 819, 924, 984, 992, 1001, 1170, 1316, 1320, 1365, 1431, 1780, 2016, 2184, 2295, 2296, 2299, 2464, 2466, 2655, 2832, 3100, 3344, 3420, 3627, 3724, 3948, 4320, 4336, 4416, 4680

Offset: 1

Author

David Consiglio, Jr., Nov 20 2014

Keywords

• nonn
• easy

Comments

Equal to the indices of the zero terms that correspond to composite numbers in A191906.

Examples

			12 is on the list because the proper divisors of 12 are [1,2,3,4,6]. The product of these numbers is 144. Their sum is 16. 144 is divisible by 16.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..1203

Crossrefs

Cf. A145551.

Programs

Maple

filter:= proc(n)
       local d,p,s;
     if isprime(n) then return false fi;
     d:= numtheory:-divisors(n) minus {n};
     convert(d,`*`) mod convert(d,`+`) = 0;
end proc:
select(filter, [$2..10000]); # Robert Israel, Dec 16 2014

Mathematica

a247145[n_Integer] :=
Select[Select[Range[n], CompositeQ[#] &],
Divisible[Times @@ Most@Divisors[#], Plus @@ Most@Divisors[#]] &]; a247145[4680] (* Michael De Vlieger, Dec 15 2014 *)
fQ[n_Integer] := Block[{d = Most@Divisors@n}, Mod[Times @@ d, Plus @@ d] == 0]; Select[Range@4680, ! PrimeQ@# && fQ@# &] (* Michael De Vlieger, Dec 19 2014, suggested by Robert G. Wilson v *)

PARI

forcomposite(n=1,10^3,d=divisors(n);p=prod(i=1,#d-1,d[i]);if(!(p%(sigma(n)-n)),print1(n,", "))) \\ Derek Orr, Nov 27 2014

Python

from functools import reduce
from operator import mul
def divs(n):
    for i in range(1, int(n / 2 + 1)):
        if n % i == 0:
            yield i
    yield n
g = []
for a in range(2, 100):
    q = list(divs(a))[0:-1]
    if reduce(mul, q, 1) % sum(q) == 0 and len(q) != 1:
        g.append(a)
print(g)

Extensions

More terms from Derek Orr, Dec 03 2014