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A083209 Numbers with exactly one subset of their sets of divisors such that the complement has the same sum. 5
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

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

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

EXAMPLE

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

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)<n then 0 else ll:= subsop(m=NULL, l); b(n, ll) +b(n-l[m], ll) fi end: a:= proc(n) option remember; local i, k, l, m, r; for k from `if`(n=1, 1, a(n-1)+1) do l:= sort([divisors(k)[]]); m:= iquo(add(i, i=l), 2, 'r'); if r=0 and b(m, l)=2 then break fi od; k end: seq(a(n), n=1..30); # 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]<n, 0, True, ll = ReplacePart[l, m -> 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-Fran├žois Alcover, Jan 31 2017, after Alois P. Heinz *)

CROSSREFS

Cf. A005101, A005835, A064771.

Sequence in context: A189793 A079760 A109895 * A080714 A116368 A290467

Adjacent sequences:  A083206 A083207 A083208 * A083210 A083211 A083212

KEYWORD

nonn

AUTHOR

Reinhard Zumkeller, Apr 22 2003

EXTENSIONS

More terms from Alois P. Heinz, Oct 29 2009

STATUS

approved

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Last modified April 26 08:35 EDT 2019. Contains 322472 sequences. (Running on oeis4.)