A364900 The n-volume of the unit regular n-simplex is sqrt(a(n))/A364901(n), with a(n) being squarefree.

1, 1, 3, 2, 5, 3, 7, 1, 1, 5, 11, 6, 13, 7, 15, 2, 17, 1, 19, 10, 21, 11, 23, 3, 1, 13, 3, 14, 29, 15, 31, 1, 33, 17, 35, 2, 37, 19, 39, 5, 41, 21, 43, 22, 5, 23, 47, 6, 1, 1, 51, 26, 53, 3, 55, 7, 57, 29, 59, 30, 61, 31, 7, 2, 65, 33, 67, 34, 69, 35, 71, 1, 73, 37, 3

Offset

0,3

Comments

a(n) = 1 if and only if n = 2*k^2 - 1 or n = 4*k^2 - 4*k for k >= 1.

a(n) = a(n+1) = 1 if and only if n = A001333(k)^2 - 2 for even k and A001333(k)^2 - 1 for odd k.

Links

Jianing Song, Table of n, a(n) for n = 0..10000

Wikipedia, Simplex.

Formula

The n-volume of the unit regular n-simplex is sqrt(n+1)/(n!*2^(n/2)), so a(n) = A007913(n+1) for even n and A007913((n+1)/2) for odd n.

Example

  n |  the n-volume of the
    | unit regular n-simplex
  2 |  sqrt(3)/4 = A120011
  3 |  sqrt(2)/12 = A020829
  4 |  sqrt(5)/96 = A364895
  5 |  sqrt(3)/480
  6 |  sqrt(7)/5760
  7 |        1/20160
  8 |        1/215040
  9 |  sqrt(5)/5806080

Mathematica

f[p_, e_] := p^Mod[e, 2]; a[0] = a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[If[EvenQ[n], n+1, (n+1)/2]]; Array[a, 75, 0] (* Amiram Eldar, Mar 18 2026 *)

Prog

(PARI) a(n) = if(n%2, core((n+1)/2), core(n+1))

Crossrefs

Cf. A007913, A364901, A120011, A020829, A364895, A001333.

Sequence in context: A129231 A166477 A124332 * A295311 A246416 A165342

Adjacent sequences: A364897 A364898 A364899 * A364901 A364902 A364903

Keyword

nonn,easy

Author

Jianing Song, Aug 12 2023

Status

approved