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Numbers k such that the sum of the anti-divisors of k exceeds 2*k.
1

%I #24 Jan 12 2020 09:55:25

%S 143,175,203,247,248,270,280,297,315,325,333,347,357,368,410,423,462,

%T 472,473,500,518,522,553,563,567,578,585,598,630,637,675,682,693,697,

%U 725,742,760,770,787,788,808,833,850,858,878,893,913,945,963,977,990

%N Numbers k such that the sum of the anti-divisors of k exceeds 2*k.

%C See A066272 for definition of anti-divisor.

%H Amiram Eldar, <a href="/A073954/b073954.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..576 from Michael De Vlieger)

%e a(1) = 143 because A066417(143) = 292, which exceeds 2 * 143 = 286. (A066417 is the sum of anti-divisors of n).

%t a066417[n_Integer] := Total[Cases[Range[2, n - 1], _?(Abs[Mod[n, #] - #/2] < 1 &)]]; Select[Range[10000], If[a066417[#] > 2 #, True, False] &] (* _Michael De Vlieger_, Aug 09 2014 *)

%o (Python)

%o A073954 = [n for n in range(1,10**5) if sum([d for d in range(2,n,2) if n%d and not 2*n%d])+sum([d for d in range(3,n,2) if n%d and 2*n%d in [d-1,1]]) > 2*n] # _Chai Wah Wu_, Aug 09 2014

%Y Cf. A066272 (number of anti-divisors of n), A066417 (sum of anti-divisors of n).

%K easy,nonn

%O 1,1

%A _Jason Earls_, Sep 03 2002