A113909 Odd numbers which are neither squares nor one less than squares.

5, 7, 11, 13, 17, 19, 21, 23, 27, 29, 31, 33, 37, 39, 41, 43, 45, 47, 51, 53, 55, 57, 59, 61, 65, 67, 69, 71, 73, 75, 77, 79, 83, 85, 87, 89, 91, 93, 95, 97, 101, 103, 105, 107, 109, 111, 113, 115, 117, 119, 123, 125, 127, 129, 131, 133, 135, 137, 139, 141, 145, 147, 149, 151, 153, 155

Offset

1,1

Comments

All such numbers may be written as n*(n+1)-k or n*(n+1)+k with k odd < n and n > 1.

The sequence may be constructed by reading the infinite square array A(n,k) by upward antidiagonals where A(n,k) = (n+k-1)*(n+k)+2*k-1 for k < n and A(n,k) = (n+k)*(n+k+1)-2*n+1 for k >= n.

Links

Paolo Xausa, Table of n, a(n) for n = 1..10000

Example

As an array A(n,k) read by upward antidiagonals:
   5 11 19 29  41 ...
   7 17 27 39  53 ...
  13 23 37 51  67 ...
  21 33 47 65  83 ...
  31 45 61 79 101 ...
  ...

Mathematica

Select[Range[1, 181, 2], NoneTrue[Sqrt[{#, # + 1}], IntegerQ] &] (* or, as an array: *)
A113909[n_, k_] := If[k < n, #*(# - 1) + 2*k - 1, #*(# + 1) - 2*n + 1] & [n + k];
Table[A113909[n - k + 1, k], {n, 12}, {k, n}] (* Paolo Xausa, Aug 22 2025 *)

Prog

(PARI) isok(n)= {n%2 && !issquare(n) && !issquare(n+1)} \\ Andrew Howroyd, Aug 17 2025

Crossrefs

Subsequence of A088828.

Sequence in context: A384321 A384322 A256387 * A391927 A391938 A111906

Adjacent sequences: A113906 A113907 A113908 * A113910 A113911 A113912

Keyword

nonn

Author

Pierre CAMI, Jan 29 2006

Extensions

Edited by N. J. A. Sloane, Jul 18 2009

Edited by Andrew Howroyd, Aug 17 2025

Status

approved