%I #27 Mar 24 2023 02:50:42

%S 1,3,5,7,11,15,17,19,23,27,29,31,35,39,41,47,53,55,59,63,65,69,71,77,

%T 79,83,87,89,95,99,103,107,111,119,125,127,131,139,143,149,155,159,

%U 161,167,175,179,191,195,197,199,203,207,209,215,219,223,227,233,239

%N Numbers k having partitions into distinct divisors of k + 1.

%C A085491(a(n)) > 0; complement of A085492.

%H Alois P. Heinz, <a href="/A085493/b085493.txt">Table of n, a(n) for n = 1..10000</a>

%H Paul K. Stockmeyer, <a href="https://doi.org/10.4169/mathhorizons.21.1.8">Of camels, inheritance, and unit fractions</a>, Math Horizons, 21 (2013), 8-11.

%F {k > 0 : 0 < [x^k] Product_{d divides (k+1)} (1+x^d)}. - _Alois P. Heinz_, Feb 04 2023

%e The divisors of 42 are 1, 2, 3, 6, 7, 14, 21, 42. Since 6 + 14 + 21 = 41, 41 is in the sequence.

%e The divisors of 43 are 1, 43. Since no selection of these divisors can possibly add up to 42, this means that 42 is not in the sequence.

%p q:= proc(m) option remember; local b, l; b, l:=

%p proc(n, i) option remember; n=0 or i>=1 and

%p (l[i]<=n and b(n-l[i], i-1) or b(n, i-1))

%p end, sort([numtheory[divisors](m+1)[]]);

%p b(m, nops(l)-1)

%p end:

%p select(q, [$1..300])[]; # _Alois P. Heinz_, Feb 04 2023

%t divNextableQ[n_] := TrueQ[Length[Select[Subsets[Divisors[n + 1]], Plus@@# == n &]] > 0]; Select[Range[100], divNextableQ] (* _Alonso del Arte_, Jan 07 2023 *)

%o (Scala) def divisors(n: Int): IndexedSeq[Int] = (1 to n).filter(n % _ == 0)

%o def divPartSums(n: Int): List[Int] = divisors(n).toSet.subsets.toList.map(_.sum)

%o (1 to 128).filter(n => divPartSums(n + 1).contains(n)) // _Alonso del Arte_, Jan 26 2023

%Y Cf. A085498, A106431.

%K nonn

%O 1,2

%A _Reinhard Zumkeller_, Jul 03 2003