A344351 Number of squarefree numbers along the main diagonal of an n X n square array whose elements are the numbers from 1..n^2, listed in increasing order by rows.

1, 1, 2, 3, 4, 4, 4, 6, 8, 6, 8, 8, 11, 9, 12, 10, 16, 12, 15, 14, 17, 14, 16, 16, 21, 18, 23, 16, 26, 18, 25, 22, 28, 21, 31, 23, 29, 26, 30, 26, 37, 27, 34, 31, 37, 28, 39, 30, 43, 36, 42, 31, 49, 35, 43, 39, 47, 35, 52, 34, 49, 43, 52, 41, 59, 40, 58, 47, 58, 44, 62, 44, 60

Offset

1,3

Comments

Number of squarefree numbers along the main diagonal of an n X n square array whose elements are the numbers from 1..n^2, listed in increasing order by rows.

Links

Table of n, a(n) for n=1..73.

Formula

a(n) = Sum_{k=1..n} mu(n*k-n+k)^2, where mu is the Möbius function.

Example

                                                      [1   2  3  4  5]
                                      [1   2  3  4]   [6   7  8  9 10]
                            [1 2 3]   [5   6  7  8]   [11 12 13 14 15]
                   [1 2]    [4 5 6]   [9  10 11 12]   [16 17 18 19 20]
           [1]     [3 4]    [7 8 9]   [13 14 15 16]   [21 22 23 24 25]
------------------------------------------------------------------------
  n         1        2         3            4                 5
------------------------------------------------------------------------
  a(n)      1        1         2            3                 4
------------------------------------------------------------------------
  numbers  {1}      {1}      {1,5}       {1,6,11}        {1,7,13,19}
------------------------------------------------------------------------

Mathematica

Table[Sum[MoebiusMu[n*(k - 1) + k]^2, {k, n}], {n, 100}]

Crossrefs

Cf. A008683 (Möbius), A344350.

Sequence in context: A339363 A377910 A114212 * A108355 A346049 A057951

Adjacent sequences: A344348 A344349 A344350 * A344352 A344353 A344354

Keyword

nonn

Author

Wesley Ivan Hurt, May 15 2021

Status

approved