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A247145
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Composite numbers such that the product of the number's proper divisors is divisible by the sum of the number's proper divisors.
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1
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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
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OFFSET
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1,1
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COMMENTS
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Equal to the indices of the zero terms that correspond to composite numbers in A191906.
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LINKS
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EXAMPLE
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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.
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MAPLE
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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:
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MATHEMATICA
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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 *)
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PROG
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(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)
(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
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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