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A083209 Numbers with exactly one subset of their sets of divisors such that the complement has the same sum. 11


%S 6,12,20,28,56,70,88,104,176,208,272,304,368,464,496,550,650,736,836,

%T 928,992,1184,1312,1376,1504,1696,1888,1952,2752,3008,3230,3392,3770,

%U 3776,3904,4030,4288,4510,4544,4672,5056,5170,5312,5696,5830,6208,6464

%N Numbers with exactly one subset of their sets of divisors such that the complement has the same sum.

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

%C 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

%C 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

%H T. D. Noe, <a href="/A083209/b083209.txt">Table of n, a(n) for n=1..407</a> (terms < 10^6)

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PerfectNumber.html">Perfect Number.</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/WeirdNumber.html">Weird Number.</a>

%H Reinhard Zumkeller, <a href="/A083206/a083206.txt">Illustration of initial terms</a>

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

%p 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

%t 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_ *)

%Y Cf. A005101, A005835, A064771.

%K nonn

%O 1,1

%A _Reinhard Zumkeller_, Apr 22 2003

%E More terms from _Alois P. Heinz_, Oct 29 2009

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Last modified June 6 01:16 EDT 2020. Contains 334858 sequences. (Running on oeis4.)