OFFSET
1,1
COMMENTS
Equivalently, abundant numbers with odd sum of divisors.
Seems to be a proper subset of A083211. - Robert G. Wilson v, Mar 30 2010. This sequence is indeed a proper subset of A083211, since the abundance of a number k, A033880(k) = sigma(k) - 2*k, has the same parity as sigma(k). If sigma(k) is odd then the sums of any two complementary subsets of the divisors of k have different parities and thus they cannot be equal. - Amiram Eldar, Jun 20 2020
If n is present, so is 2*n. - Robert G. Wilson v, Jun 21 2015
If n is in the sequence, so is 100*n (conjectured). - Sergey Pavlov, Mar 22 2017. Pavlov's observation trivially follows from the fact that to have odd abundance a number k must be either a square or twice a square. If such a number k is abundant then 100*k = (10^2) * k is abundant as well and has odd abundance. In general, we can say that if k is present, so are t^2*k and 2*t^2*k, for every t>0. - Giovanni Resta, Oct 16 2018
Terms are congruent to {0, 2, 4, 8, 9, 14, 16, 18, 20, 26, 28, 32} (mod 36). - Robert G. Wilson v, Dec 09 2018
LINKS
Robert G. Wilson v, Table of n, a(n) for n = 1..22927 (corrected by Michel Marcus)
Eric Weisstein's World of Mathematics, Abundant Number
EXAMPLE
k = 18 is in the sequence because its divisors are {1,2,3,6,9,18} which sum to sigma(k) = 39; so its abundance is sigma(k) - 2k = 39 - 36 = 3.
MAPLE
with(numtheory): select(k->sigma(k)>2*k and modp(sigma(k)-2*k, 2)=1, [$1..5000]); # Muniru A Asiru, Dec 11 2018
MATHEMATICA
abundance[n_] := DivisorSigma[1, n] - 2 n; Select[Range[1000], abundance[#] > 0 && Mod[abundance[#], 2] == 1 &]
abundOddAbundQ[n_] := If[MemberQ[{0, 2, 4, 8, 9, 14, 16, 18, 20, 26, 28, 32}, Mod[n, 36]], a = DivisorSigma[1, n]; OddQ@a && a > 2 n]; Select[ Range@ 5000, abundOddAbundQ@# &] (* Robert G. Wilson v, Dec 23 2018 *)
PROG
(PARI) is(n)=my(k=sigma(n)-2*n); k>0 && k%2 \\ Charles R Greathouse IV, Feb 21 2017
(Python)
from sympy.ntheory import divisor_sigma
def a(n):
return divisor_sigma(n) - 2*n
[n for n in range(18, 5001) if a(n) > 0 and a(n) % 2] # Indranil Ghosh, Mar 22 2017
(GAP) Filtered([1..5000], k->Sigma(k)-2*k>0 and (Sigma(k)-2*k) mod 2=1); # Muniru A Asiru, Dec 11 2018
CROSSREFS
KEYWORD
nonn
AUTHOR
Robert G. Wilson v, Feb 17 2009
EXTENSIONS
Name edited by Michel Marcus and Charles R Greathouse IV, Mar 26 2017
Edited by N. J. A. Sloane, Jun 21 2020 at the suggestion of Amiram Eldar
STATUS
approved