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A083209 Numbers with exactly one subset of their sets of divisors such that the complement has the same sum. 13
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 * A339858 A080714 A116368

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 13 01:36 EDT 2021. Contains 342934 sequences. (Running on oeis4.)