A348388 - OEIS

%I #24 Nov 06 2021 11:08:05

%S 1,2,3,1,4,1,5,2,1,6,2,1,7,3,1,1,8,3,2,1,9,4,2,1,1,10,4,2,1,1,11,5,3,

%T 2,1,1,12,5,3,2,1,1,13,6,3,2,1,1,1,14,6,4,2,2,1,1,15,7,4,3,2,1,1,1,16,

%U 7,4,3,2,1,1,1,17,8,5,3,2,2,1,1,1,18,8,5,3,2,2,1,1,1,19,9,5,4,3,2,1,1,1,1

%N Irregular triangle read by rows: T(n, k) = floor((n-k)/k), for k = 1, 2, ..., floor(n/2) and n >= 2.

%C This irregular triangle T(n, k) gives the number of multiples of number k, larger than k and not exceeding n, for k = 1, 2, ..., floor(n/2), for n >= 2. See A348389 for the array of these multiples.

%C The length of row n is floor(n/2) = A004526(n), for n >= 2.

%C The row sums give A002541(n). See the formula given there by _Wesley Ivan Hurt_, May 08 2016.

%C The columns give the k-fold repeated positive integers k, for k >= 1.

%H Karl-Heinz Hofmann, <a href="/A348388/b348388.txt">Table of n, a(n) for n = 2..10101</a>

%F T(n, k) =  floor((n-k)/k), for k = 1, 2, ..., floor(n/2) and n >= 2.

%F G.f. of column k: G(k, x) = x^(2*k)/((1 - x)*(1 - x^k)).

%e The irregular triangle T(n, k) begins:

%e n\k   1 2 3 4 5 6 7 8 9 10 ...

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

%e 2:    1

%e 3:    2

%e 4:    3 1

%e 5:    4 1

%e 6:    5 2 1

%e 7:    6 2 1

%e 8:    7 3 1 1

%e 9:    8 3 2 1

%e 10:   9 4 2 1 1

%e 11:  10 4 2 1 1

%e 12:  11 5 3 2 1 1

%e 13:  12 5 3 2 1 1

%e 14:  13 6 3 2 1 1 1

%e 15:  14 6 4 2 2 1 1

%e 16:  15 7 4 3 2 1 1 1

%e 17:  16 7 4 3 2 1 1 1

%e 18:  17 8 5 3 2 2 1 1 1

%e 19:  18 8 5 3 2 2 1 1 1

%e 20:  19 9 5 4 3 2 1 1 1  1

%e ...

%t T[n_, k_] := Floor[(n - k)/k]; Table[T[n, k], {n, 2, 20}, {k, 1, Floor[n/2]}] // Flatten (* _Amiram Eldar_, Nov 02 2021 *)

%o (Python)

%o def A348388row(n): return [(n - k) // k for k in range(1, 1 + n // 2)]

%o for n in range(2, 21): print(A348388row(n))  # _Peter Luschny_, Nov 05 2021

%Y Columns k (with varying offsets): A000027, A004526, A008620, A008621, A002266, A097992, ...

%Y Cf. A002541, A004526, A010766, A348389.

%K nonn,easy,tabf

%O 2,2

%A _Wolfdieter Lang_, Oct 31 2021