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%I #86 Jul 31 2021 23:00:03
%S 18,36,72,100,144,162,196,200,288,324,392,400,450,576,648,784,800,882,
%T 900,968,1152,1296,1352,1458,1568,1600,1764,1800,1936,2178,2304,2450,
%U 2500,2592,2704,2916,3042,3136,3200,3528,3600,3872,4050,4356,4608,4624
%N Abundant numbers (A005101) whose abundance is odd.
%C Equivalently, abundant numbers with odd sum of divisors.
%C Complement of A204825 with respect to A005101 (abundant numbers).
%C 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
%C If n is present, so is 2*n. - _Robert G. Wilson v_, Jun 21 2015
%C 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
%C 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
%H Robert G. Wilson v, <a href="/A156903/b156903.txt">Table of n, a(n) for n = 1..22927</a> (corrected by Michel Marcus)
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/AbundantNumber.html">Abundant Number</a>
%e 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.
%p with(numtheory): select(k->sigma(k)>2*k and modp(sigma(k)-2*k,2)=1,[$1..5000]); # _Muniru A Asiru_, Dec 11 2018
%t abundance[n_] := DivisorSigma[1, n] - 2 n; Select[Range[1000], abundance[#] > 0 && Mod[abundance[#], 2] == 1 &]
%t 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 *)
%o (PARI) is(n)=my(k=sigma(n)-2*n); k>0 && k%2 \\ _Charles R Greathouse IV_, Feb 21 2017
%o (Python)
%o from sympy.ntheory import divisor_sigma
%o def a(n):
%o return divisor_sigma(n) - 2*n
%o [n for n in range(18, 5001) if a(n) > 0 and a(n) % 2] # _Indranil Ghosh_, Mar 22 2017
%o (GAP) Filtered([1..5000],k->Sigma(k)-2*k>0 and (Sigma(k)-2*k) mod 2=1); # _Muniru A Asiru_, Dec 11 2018
%Y Intersection of A005101 and A028982. - _Amiram Eldar_, Jun 20 2020
%Y Cf. A000203, A033880, A259231. A proper subset of A083211.
%Y Cf. A204825 (abundant numbers with even sum of divisors), A204826 (deficient numbers with odd sum of divisors), A204827 (deficient numbers with even sum of divisors).
%K nonn
%O 1,1
%A _Robert G. Wilson v_, Feb 17 2009
%E Name edited by _Michel Marcus_ and _Charles R Greathouse IV_, Mar 26 2017
%E Edited by _N. J. A. Sloane_, Jun 21 2020 at the suggestion of _Amiram Eldar_
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