A143057 A positive integer n is included if, for at least one divisor k of n and for at least one divisor j of (n+1), (k+1)*(n/k +1) = (j+1)*((n+1)/j +1).

5, 14, 24, 27, 44, 55, 65, 90, 98, 119, 120, 152, 153, 189, 209, 220, 230, 275, 299, 322, 324, 360, 377, 390, 434, 459, 493, 495, 551, 560, 608, 620, 629, 702, 735, 779, 782, 805, 840, 860, 874, 945, 1014, 1025, 1034, 1053, 1127, 1188, 1189, 1224, 1247, 1325

Offset

1,1

Comments

Given positive integers k, d with d | k+1, then n = k*(k+d+1+(k+1)/d) is a term, with the given k and j = k+d. - Robert Israel, Apr 22 2021

Links

Robert Israel, Table of n, a(n) for n = 1..10000

Example

Let n = 24. Then (k+1)*(n/k +1), when k is taken over the divisors of 24, has the values: 50, 39, 36, 35, 35, 36, 39, 50. And (j+1)*((n+1)/j +1), when j is taken over the divisors of 25, has the values: 52, 36, 52.
Since 36 occurs in both lists of products, 24 is in sequence A143057.

Maple

filter:= proc(n) local k, a, j, r, q, d;
  for k in numtheory:-divisors(n) do
    a:= (k+1)*(n/k+1);
    r:= (a-n)^2-4*a;
    if issqr(r) then
      q:= sqrt(r)/2;
      for d in [a/2-n/2-1+q, a/2-n/2-1-q] do
        if d::posint and n+1 mod d = 0 then return true fi
      od
    fi
  od;
  false
end proc:
select(filter, [$1..10000]); # Robert Israel, Apr 22 2021

Mathematica

okQ[n_] := Module[{k, j}, k = Divisors[n]; j = Divisors[n+1]; Intersection[(k+1)(n/k+1), (j+1)((n+1)/j+1)] != {}];
Select[Range[2000], okQ] (* Jean-François Alcover, May 16 2023 *)

Crossrefs

Sequence in context: A044057 A147825 A174263 * A063238 A063223 A070722

Adjacent sequences: A143054 A143055 A143056 * A143058 A143059 A143060

Keyword

nonn

Author

Leroy Quet, Jul 20 2008

Extensions

Extended by R. J. Mathar, Aug 14 2008

Status

approved