cp's OEIS Frontend

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

%I A085493 #27 Mar 24 2023 02:50:42
%S A085493 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 A085493 79,83,87,89,95,99,103,107,111,119,125,127,131,139,143,149,155,159,
%U A085493 161,167,175,179,191,195,197,199,203,207,209,215,219,223,227,233,239
%N A085493 Numbers k having partitions into distinct divisors of k + 1.
%C A085493 A085491(a(n)) > 0; complement of A085492.
%H A085493 Alois P. Heinz, <a href="/A085493/b085493.txt">Table of n, a(n) for n = 1..10000</a>
%H A085493 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 A085493 {k > 0 : 0 < [x^k] Product_{d divides (k+1)} (1+x^d)}. - _Alois P. Heinz_, Feb 04 2023
%e A085493 The divisors of 42 are 1, 2, 3, 6, 7, 14, 21, 42. Since 6 + 14 + 21 = 41, 41 is in the sequence.
%e A085493 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 A085493 q:= proc(m) option remember; local b, l; b, l:=
%p A085493       proc(n, i) option remember; n=0 or i>=1 and
%p A085493         (l[i]<=n and b(n-l[i], i-1) or b(n, i-1))
%p A085493       end, sort([numtheory[divisors](m+1)[]]);
%p A085493       b(m, nops(l)-1)
%p A085493     end:
%p A085493 select(q, [$1..300])[];  # _Alois P. Heinz_, Feb 04 2023
%t A085493 divNextableQ[n_] := TrueQ[Length[Select[Subsets[Divisors[n + 1]], Plus@@# == n &]] > 0]; Select[Range[100], divNextableQ] (* _Alonso del Arte_, Jan 07 2023 *)
%o A085493 (Scala) def divisors(n: Int): IndexedSeq[Int] = (1 to n).filter(n % _ == 0)
%o A085493 def divPartSums(n: Int): List[Int] = divisors(n).toSet.subsets.toList.map(_.sum)
%o A085493 (1 to 128).filter(n => divPartSums(n + 1).contains(n)) // _Alonso del Arte_, Jan 26 2023
%Y A085493 Cf. A085498, A106431.
%K A085493 nonn
%O A085493 1,2
%A A085493 _Reinhard Zumkeller_, Jul 03 2003