A085493 Numbers k having partitions into distinct divisors of k + 1.

1, 3, 5, 7, 11, 15, 17, 19, 23, 27, 29, 31, 35, 39, 41, 47, 53, 55, 59, 63, 65, 69, 71, 77, 79, 83, 87, 89, 95, 99, 103, 107, 111, 119, 125, 127, 131, 139, 143, 149, 155, 159, 161, 167, 175, 179, 191, 195, 197, 199, 203, 207, 209, 215, 219, 223, 227, 233, 239

Offset: 1

Author

Reinhard Zumkeller, Jul 03 2003

Keywords

• nonn

Comments

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

Examples

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

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..10000
• Paul K. Stockmeyer, Of camels, inheritance, and unit fractions, Math Horizons, 21 (2013), 8-11.

Crossrefs

Cf. A085498, A106431.

Programs

Maple

q:= proc(m) option remember; local b, l; b, l:=
      proc(n, i) option remember; n=0 or i>=1 and
        (l[i]<=n and b(n-l[i], i-1) or b(n, i-1))
      end, sort([numtheory[divisors](m+1)[]]);
      b(m, nops(l)-1)
    end:
select(q, [$1..300])[];  # Alois P. Heinz, Feb 04 2023

Mathematica

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

Scala

def divisors(n: Int): IndexedSeq[Int] = (1 to n).filter(n % _ == 0)
def divPartSums(n: Int): List[Int] = divisors(n).toSet.subsets.toList.map(_.sum)
(1 to 128).filter(n => divPartSums(n + 1).contains(n)) // Alonso del Arte, Jan 26 2023

Formula

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