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

18, 36, 72, 100, 144, 162, 196, 200, 288, 324, 392, 400, 450, 576, 648, 738, 748, 774, 784, 800, 846, 882, 900, 954, 968, 1062, 1098, 1152, 1206, 1278, 1296, 1314, 1352, 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

Offset: 1

Author

Reinhard Zumkeller, Apr 22 2003

Keywords

• nonn
• changed

Comments

A083206(a(n)) = 0; subsequence of A083210.

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

Examples

			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.
		

Links

• David A. Corneth, Table of n, a(n) for n = 1..12915 (first 6061 terms from Antti Karttunen)
• Eric Weisstein's World of Mathematics, Abundant Number.
• Reinhard Zumkeller, Illustration of initial terms

Crossrefs

Intersection of A005101 and A083210.

Disjoint union of A156903 and A171641. - Amiram Eldar, Jun 20 2020

Positions of negative terms in A378600.

Cf. A000203, A028982, A083206, A156942 (odd terms), A378661 (characteristic function).

Programs

Mathematica

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 *)

PARI

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)); };
is_A083211(n) = ((sigma(n)>2*n) && (0==A083206(n))); \\ Antti Karttunen, Dec 04 2024

Formula

{k such that sigma(k) > 2*k and A083206(k) = 0}. - Antti Karttunen, Dec 04 2024

Extensions

a(21)-a(46) from Robert G. Wilson v, Apr 01 2010

Many missing terms inserted, first ones at a(29) = 1206 and a(30) = 1278 - Antti Karttunen, Dec 04 2024