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

1,2

Comments

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

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.

Formula

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

Example

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.

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 *)

Prog

(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

Crossrefs

Cf. A085498, A106431.

Sequence in context: A356065 A082418 A319899 * A343621 A139252 A076245

Adjacent sequences: A085490 A085491 A085492 * A085494 A085495 A085496

Keyword

nonn

Author

Reinhard Zumkeller, Jul 03 2003

Status

approved