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

%I A083209 #41 Feb 16 2025 08:32:49
%S A083209 6,12,20,28,56,70,88,104,176,208,272,304,368,464,496,550,650,736,836,
%T A083209 928,992,1184,1312,1376,1504,1696,1888,1952,2752,3008,3230,3392,3770,
%U A083209 3776,3904,4030,4288,4510,4544,4672,5056,5170,5312,5696,5830,6208,6464
%N A083209 Numbers whose divisors can be partitioned in exactly one way into two disjoint sets with the same sum.
%C A083209 A083206(a(n))=1; perfect numbers (A000396) are a subset; problem: are weird numbers (A006037) a subset?
%C A083209 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 A083209 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
%C A083209 "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
%H A083209 T. D. Noe, <a href="/A083209/b083209.txt">Table of n, a(n) for n=1..407</a> (terms < 10^6)
%H A083209 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PerfectNumber.html">Perfect Number.</a>
%H A083209 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/WeirdNumber.html">Weird Number.</a>
%H A083209 Reinhard Zumkeller, <a href="/A083206/a083206.txt">Illustration of initial terms</a>
%H A083209 <a href="/index/O#opnseqs">Index entries for sequences where odd perfect numbers must occur, if they exist at all</a>
%e A083209 n=20: 2+4+5+10 = 1+20, 20 is a term (A083206(20)=1).
%p A083209 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 A083209 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-François Alcover_, Jan 31 2017, after _Alois P. Heinz_ *)
%o A083209 (PARI) isA083209 = A378449; \\ _Antti Karttunen_, Nov 28 2024
%Y A083209 Subsequence of A083207, Zumkeller numbers.
%Y A083209 Positions of 1's in A083206.
%Y A083209 Cf. A005101, A005835, A064771, A337739 (terms with record number of divisors), A378449 (characteristic function), A378530 (subsequence).
%Y A083209 Cf. also A378652, and A335143, A335199, A335202, A335219, A335217, A339980 for variants.
%K A083209 nonn
%O A083209 1,1
%A A083209 _Reinhard Zumkeller_, Apr 22 2003
%E A083209 More terms from _Alois P. Heinz_, Oct 29 2009
%E A083209 Improved the definition, old name moved to the comments - _Antti Karttunen_, Dec 02 2024