cp's OEIS Frontend

A140863 Odd numbers k such that sigma(m) = 2m+k has a solution in m, where sigma is the sum-of-divisors function A000203.

%I A140863 #38 Mar 10 2025 16:39:45
%S A140863 3,7,17,19,31,39,41,51,59,65,71,89,115,119,127,161,185,199,215,243,
%T A140863 251,259,265,269,299,309,353,363,399,401,455,459,467,499,519,593,635,
%U A140863 713,737,815,831,845,899,921,923,965,967,983,1011,1021,1025,1049,1053,1055
%N A140863 Odd numbers k such that sigma(m) = 2m+k has a solution in m, where sigma is the sum-of-divisors function A000203.
%C A140863 From _M. F. Hasler_ and _Farideh Firoozbakht_, Nov 26 2009: (Start)
%C A140863 The sequence of Mersenne primes, A000668 is a subsequence of this sequence.
%C A140863 Because if k=2^p-1 is prime then n=2^(p-1)*(2^p-1)^2 is a solution of the equation sigma(x)=2x+k. The proof is easy. (End)
%C A140863 The definition is equivalent to asking for a number m with abundance A033880(m) = k. - _M. F. Hasler_, Mar 10 2025
%D A140863 J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 196, p. 58, Ellipses, Paris 2008.
%H A140863 Robert G. Wilson v, <a href="/A140863/b140863.txt">Table of n, a(n) for n = 1..579</a>
%H A140863 Farideh Firoozbakht and M. F. Hasler, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL13/Hasler/hasler2.html">Variations on Euclid's formula for Perfect Numbers</a>, JIS 13 (2010) #10.3.1.
%F A140863 A033880(A156903), image of A156903 under A033880, or range of A033880 restricted to A156903, where A033880 is the abundance sigma(x)-2x, and A156903 are numbers with odd positive abundance. - _M. F. Hasler_, Mar 10 2025
%Y A140863 Cf. A000668. - _M. F. Hasler_ and _Farideh Firoozbakht_, Nov 26 2009
%Y A140863 Cf. A156903. - _Robert G. Wilson v_, Dec 09 2018
%Y A140863 Cf. A000203 (sigma), A033880 (abundance: sigma(n)-2n).
%Y A140863 Cf. A380866 (smallest solutions m to the given equation).
%K A140863 nonn
%O A140863 1,1
%A A140863 _Lekraj Beedassy_, Jul 20 2008
%E A140863 a(13)-a(54) from _Donovan Johnson_, Dec 09 2008