A083209 Numbers whose divisors can be partitioned in exactly one way into two disjoint sets with the same sum.

6, 12, 20, 28, 56, 70, 88, 104, 176, 208, 272, 304, 368, 464, 496, 550, 650, 736, 836, 928, 992, 1184, 1312, 1376, 1504, 1696, 1888, 1952, 2752, 3008, 3230, 3392, 3770, 3776, 3904, 4030, 4288, 4510, 4544, 4672, 5056, 5170, 5312, 5696, 5830, 6208, 6464

Offset: 1

Author

Reinhard Zumkeller, Apr 22 2003

Keywords

• nonn

Comments

A083206(a(n))=1; perfect numbers (A000396) are a subset; problem: are weird numbers (A006037) a subset?

The weird numbers A006037 are not a subset of this sequence. The first missing weird number is A006037(8) = 10430. - Alois P. Heinz, Oct 29 2009

All numbers of the form p*2^k are in this sequence for k>0 and odd primes p between 2^(k+1)/3 and 2^(k+1). - T. D. Noe, Jul 08 2010

"Numbers with exactly one subset of their sets of divisors such that the complement has the same sum." - This was the original name of the sequence, but strictly taken is incorrect, because there are always two subsets that satisfy this condition: the subset and its complement. - Antti Karttunen, Dec 02 2024

Examples

			n=20: 2+4+5+10 = 1+20, 20 is a term (A083206(20)=1).
		

Links

• T. D. Noe, Table of n, a(n) for n=1..407 (terms < 10^6)
• Eric Weisstein's World of Mathematics, Perfect Number.
• Eric Weisstein's World of Mathematics, Weird Number.
• Reinhard Zumkeller, Illustration of initial terms
• Index entries for sequences where odd perfect numbers must occur, if they exist at all

Crossrefs

Subsequence of A083207, Zumkeller numbers.

Positions of 1's in A083206.

Cf. A005101, A005835, A064771, A337739 (terms with record number of divisors), A378449 (characteristic function), A378530 (subsequence).

Cf. also A378652, and A335143, A335199, A335202, A335219, A335217, A339980 for variants.

Programs

Maple

with(numtheory): b:= proc(n,l) option remember; local m, ll, i; m:= nops(l); if n<0 then 0 elif n=0 then 1 elif m=0 or add(i, i=l)Alois P. Heinz, Oct 29 2009

Mathematica

b[n_, l_] := b[n, l] = Module[{m, ll, i}, m = Length[l]; Which[n<0, 0, n == 0, 1, m == 0 || Total[l] Nothing]; b[n, ll] + b[n - l[[m]], ll]]]; a[n_] := a[n] = Module[{i, k, l, m, r}, For[k = If[n == 1, 1, a[n-1]+1], True, k++, l = Divisors[k]; {m, r} = QuotientRemainder[Total[l], 2]; If[r==0 && b[m, l]==2, Break[]]]; k]; Table[Print["a(", n, ") = ", a[n]]; a[n], {n, 1, 50}] (* Jean-François Alcover, Jan 31 2017, after Alois P. Heinz *)

PARI

isA083209 = A378449; \\ Antti Karttunen, Nov 28 2024

Extensions

More terms from Alois P. Heinz, Oct 29 2009

Improved the definition, old name moved to the comments - Antti Karttunen, Dec 02 2024