A272890 Numbers k such that the product of k and the sum of the reciprocal of their anti-divisors is an integer.

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

Offset

1,1

Links

Chai Wah Wu and Charles R Greathouse IV, Table of n, a(n) for n = 1..450 (first 191 terms from Chai Wah Wu)

Example

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.

Maple

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);

Mathematica

Select[Range[3, 10^4], Function[n, IntegerQ[n Total@ #] &[1/Select[Range[2, n - 1], (Abs[Mod[n, #] - #/2] < 1 &)]]]](* Michael De Vlieger, May 13 2016, after Harvey P. Dale at A066272 *)

Prog

(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)))
is(n)=denominator(vecsum(apply(k->1/k, ad(n)))*n)==1 && n>2 \\ Charles R Greathouse IV, May 12 2016

Crossrefs

Cf. A066272.

Sequence in context: A262782 A204513 A223306 * A129425 A246757 A079655

Adjacent sequences: A272887 A272888 A272889 * A272891 A272892 A272893

Keyword

nonn,changed

Author

Paolo P. Lava, May 09 2016

Extensions

a(15) inserted and more terms added by Chai Wah Wu, May 10 2016

Status

approved