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A083211 Abundant numbers (A005101) with no subset of their divisors such that the complement has the same sum.

%I A083211 #52 Aug 21 2025 14:39:32
%S A083211 18,36,72,100,144,162,196,200,288,324,392,400,450,576,648,738,748,774,
%T A083211 784,800,846,882,900,954,968,1062,1098,1152,1206,1278,1296,1314,1352,
%U A083211 1422,1458,1494,1568,1600,1602,1746,1764,1800,1818,1854,1926,1936,1962,2034,2178,2286,2304,2358,2450,2466,2500,2502,2592,2682,2704,2718,2826,2916,2934,3006,3042
%N A083211 Abundant numbers (A005101) with no subset of their divisors such that the complement has the same sum.
%C A083211 A083206(a(n)) = 0; subsequence of A083210.
%C A083211 All [abundant] numbers with an odd sum of divisors (either a square or twice a square, A028982) must be terms because for these numbers the two subsets will be of opposite parity. - _Robert G. Wilson v_, Apr 01 2010, clarified by _Antti Karttunen_, Dec 05 2024
%H A083211 David A. Corneth, <a href="/A083211/b083211.txt">Table of n, a(n) for n = 1..12915</a> (first 6061 terms from Antti Karttunen)
%H A083211 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/AbundantNumber.html">Abundant Number</a>.
%H A083211 Reinhard Zumkeller, <a href="/A083206/a083206.txt">Illustration of initial terms</a>
%F A083211 {k such that sigma(k) > 2*k and A083206(k) = 0}. - _Antti Karttunen_, Dec 04 2024
%e A083211 Divisors of n=18: {1,2,3,6,9,18}; 18 is pseudo-perfect (A005835): 18=9+6+3, but there exist no two complementary subsets of divisors having the same sum, therefore 18 is a term.
%t A083211 fQ[n_] := Block[{d = Divisors[n], t, ds, x}, ds = Total[d]; If[Mod[ds, 2] > 0, False, t = CoefficientList[Product[1 + x^i, {i, d}], x]; t[[1 + ds/2]] > 0]]; Select[Range[3042], And[DivisorSigma[1, #] > 2 #, ! fQ[#]] &] (* _Michael De Vlieger_, Dec 04 2024, after _T. D. Noe_ at A083207 *)
%o A083211 (PARI)
%o A083211 A083206(n) = { my(s=sigma(n),p=1); if(s%2 || s < 2*n, 0, fordiv(n, d, p *= ('x^d + 'x^-d)); (polcoeff(p, 0)/2)); };
%o A083211 is_A083211(n) = ((sigma(n)>2*n) && (0==A083206(n))); \\ _Antti Karttunen_, Dec 04 2024
%Y A083211 Intersection of A005101 and A083210.
%Y A083211 Disjoint union of A156903 and A171641. - _Amiram Eldar_, Jun 20 2020
%Y A083211 Positions of negative terms in A378600.
%Y A083211 Cf. A000203, A028982, A083206, A156942 (odd terms), A378661 (characteristic function).
%K A083211 nonn,changed
%O A083211 1,1
%A A083211 _Reinhard Zumkeller_, Apr 22 2003
%E A083211 a(21)-a(46) from _Robert G. Wilson v_, Apr 01 2010
%E A083211 Many missing terms inserted, first ones at a(29) = 1206 and a(30) = 1278 - _Antti Karttunen_, Dec 04 2024