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A272890
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Numbers n such that the product of n and the sum of the reciprocal of their anti-divisors is an integer.
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1
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9, 36, 441, 576, 1296, 1764, 2025, 7569, 10404, 17424, 23409, 34596, 41616, 51984, 56169, 74529, 88209, 90000, 103041, 140625, 181476, 194481, 219024, 236196, 239121, 269361, 324900, 367236, 404496, 480249, 540225, 571536, 576081, 627264, 783225, 842724, 904401
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OFFSET
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1,1
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COMMENTS
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Values of these products are 6, 6, 370, 6, 60, 370, 1875, 5661, 1995, 864, 18573, 6454, 1995, 2476, 41086, 67801, 72684, 5076, 75120, ...
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LINKS
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EXAMPLE
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Anti-divisors of 9 are 2 and 6: 9 * (1/2 + 1/6) = 6;
Anti-divisors of 441 are 2, 6, 14, 18, 42, 98, 126 and 294: 441 * (1/2 + 1/6 + 1/14 + 1/18 + 1/42 + 1/98 + 1/126 + 1/294) = 370.
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MAPLE
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with(numtheory); P:=proc(q) local a, k, n; for n from 3 to q do a:=0;
for k from 2 to n-1 do if abs((n mod k)-k/2)<1 then a:=n/k+a; fi; od;
if type(a, integer) then print(n); fi; od; end: P(10^6);
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MATHEMATICA
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PROG
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(Python)
from fractions import Fraction
from sympy.ntheory.factor_ import antidivisors
A272890_list = [n for n in range(3, 10**5) if sum(Fraction(n, a) for a in antidivisors(n)).denominator == 1] # Chai Wah Wu, May 10 2016
(Ruby)
def f(n)
ary = []
(2..n).each{|i|
if i % 2 == 0
ary << i if n % i == i / 2
else
ary << i if (n % i == (i - 1) / 2) || (n % i == (i + 1) / 2)
end
}
ary
end
def g(ary)
ary.inject(0){|s, i| s + 1r / i}
end
p (3..10 ** 5).select{|i| (i * g(f(i))).denominator == 1} # Seiichi Manyama, May 12 2016
(PARI) ad(n)=select(t->n%t && t<n, concat(concat(divisors(2*n-1), divisors(2*n+1)), 2*divisors(n)))
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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a(15) inserted and more terms added by Chai Wah Wu, May 10 2016
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STATUS
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approved
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