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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.
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%I #6 May 31 2021 19:41:27

%S 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,

%T 16,26,18,25,22,28,21,31,23,29,26,30,26,37,27,34,31,37,28,39,30,43,36,

%U 42,31,49,35,43,39,47,35,52,34,49,43,52,41,59,40,58,47,58,44,62,44,60

%N 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.

%C 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.

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

%e [1 2 3 4 5]

%e [1 2 3 4] [6 7 8 9 10]

%e [1 2 3] [5 6 7 8] [11 12 13 14 15]

%e [1 2] [4 5 6] [9 10 11 12] [16 17 18 19 20]

%e [1] [3 4] [7 8 9] [13 14 15 16] [21 22 23 24 25]

%e ------------------------------------------------------------------------

%e n 1 2 3 4 5

%e ------------------------------------------------------------------------

%e a(n) 1 1 2 3 4

%e ------------------------------------------------------------------------

%e numbers {1} {1} {1,5} {1,6,11} {1,7,13,19}

%e ------------------------------------------------------------------------

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

%Y Cf. A008683 (Möbius), A344350.

%K nonn

%O 1,3

%A _Wesley Ivan Hurt_, May 15 2021