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A380600 Irregular table T(n, k), n > 0, k = 1..A000005(n) read by rows: the n-th row lists the numbers of the form n * (d-1) / d with d a positive divisor of n.

%I A380600 #21 Apr 01 2025 08:56:31
%S A380600 0,0,1,0,2,0,2,3,0,4,0,3,4,5,0,6,0,4,6,7,0,6,8,0,5,8,9,0,10,0,6,8,9,
%T A380600 10,11,0,12,0,7,12,13,0,10,12,14,0,8,12,14,15,0,16,0,9,12,15,16,17,0,
%U A380600 18,0,10,15,16,18,19,0,14,18,20,0,11,20,21,0,22
%N A380600 Irregular table T(n, k), n > 0, k = 1..A000005(n) read by rows: the n-th row lists the numbers of the form n * (d-1) / d with d a positive divisor of n.
%H A380600 <a href="/index/Di#divisors">Index entries for sequences related to divisors</a>
%F A380600 T(n, k) = n * (A027750(n, k) - 1) / A027750(n, k).
%F A380600 Sum_{k = 1..A000005(n)} T(n, k) = A094471(n).
%F A380600 Product_{k = 2..A000005(n)} T(n, k) = A072513(n).
%F A380600 LCM{k = 2..A000005(n)} T(n, k) = A258324(n).
%F A380600 T(n, 1) = 0.
%F A380600 T(n, 2) = A060681(n) for any n > 1. - _Michel Marcus_, Feb 03 2025
%F A380600 T(n, A000005(n)-1) = A046666(n) for any n > 1.
%F A380600 T(n, A000005(n)) = n-1.
%e A380600 Table T(n, k) begins:
%e A380600   n   n-th row
%e A380600   --  ------------------
%e A380600    1  0
%e A380600    2  0, 1
%e A380600    3  0, 2
%e A380600    4  0, 2, 3
%e A380600    5  0, 4
%e A380600    6  0, 3, 4, 5
%e A380600    7  0, 6
%e A380600    8  0, 4, 6, 7
%e A380600    9  0, 6, 8
%e A380600   10  0, 5, 8, 9
%e A380600   11  0, 10
%e A380600   12  0, 6, 8, 9, 10, 11
%e A380600   13  0, 12
%e A380600   14  0, 7, 12, 13
%t A380600 Table[Map[n*(# - 1)/# &, Divisors[n]], {n, 23}] // Flatten (* _Michael De Vlieger_, Feb 03 2025 *)
%o A380600 (PARI) row(n) = apply (d -> n*(d-1)/d, divisors(n))
%Y A380600 Cf. A000005, A027750, A046666, A060681, A072513, A094471 (row sums), A258324.
%K A380600 nonn,tabf,easy
%O A380600 1,5
%A A380600 _Rémy Sigrist_, Feb 02 2025