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

1,1

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

Links

David A. Corneth, Table of n, a(n) for n = 1..12915 (first 6061 terms from Antti Karttunen)

Antti Karttunen, Table of n, a(n) for n = 1..6061

Eric Weisstein's World of Mathematics, Abundant Number.

Reinhard Zumkeller, Illustration of initial terms

Formula

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

Example

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.

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 *)

Prog

(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

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, A378661 (characteristic function).

Sequence in context: A097926 A087967 A070224 * A156903 A204824 A252424

Adjacent sequences: A083208 A083209 A083210 * A083212 A083213 A083214

Keyword

nonn

Author

Reinhard Zumkeller, Apr 22 2003

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

Status

approved