A373716 a(n) is the number of distinct products i*j minus the number of distinct sums i+j with 1 <= i, j <= n.

0, 0, 1, 2, 5, 7, 12, 15, 19, 23, 32, 36, 47, 53, 60, 66, 81, 88, 105, 113, 123, 133, 154, 162, 176, 188, 201, 212, 239, 249, 278, 291, 307, 323, 341, 352, 387, 405, 424, 438, 477, 492, 533, 551, 570, 592, 637, 652, 681, 701, 726, 747, 798, 818, 847, 867, 895

Offset

1,4

Links

Table of n, a(n) for n=1..57.

Formula

a(n) = A027424(n) - A005408(n-1).

a(n) = (n-1)^2 - A062851(n).

Example

a(5) = 5 because:
 Products:                   Sums:
 * | 1 |  2 |  3 |  4 |  5   + | 1 | 2 | 3 | 4 |  5
 -------------------------   -----------------------
 1 | 1 |  2 |  3 |  4 |  5   1 | 2 | 3 | 4 | 5 |  6
 2 | 2 |  4 |  6 |  8 | 10   2 | 3 | 4 | 5 | 6 |  7
 3 | 3 |  6 |  9 | 12 | 15   3 | 4 | 5 | 6 | 7 |  8
 4 | 4 |  8 | 12 | 16 | 20   4 | 5 | 6 | 7 | 8 |  9
 5 | 5 | 10 | 15 | 20 | 25   5 | 6 | 7 | 8 | 9 | 10
The number of distinct products [1,2,3,4,5,6,8,9,10,12,15,16,20,25] is 14.
The number of distinct sums [2,3,4,5,6,7,8,9,10] is 9.
So a(5) = 14 - 9 = 5.

Prog

(Python)
A027424 = lambda n: len({i*j for i in range(1, n+1) for j in range(1, i+1)})
a = lambda n: A027424(n)-((n<<1)-1)
print([a(n) for n in range(1, 58)])
(PARI) a(n) = #setbinop((x, y)->x*y, vector(n, i, i)) - 2*n + 1; \\ Michel Marcus, Jun 23 2024

Crossrefs

Cf. A000290, A005408, A027424, A062851.

Sequence in context: A294146 A022758 A349744 * A129232 A088822 A080182

Adjacent sequences: A373713 A373714 A373715 * A373717 A373718 A373719

Keyword

nonn

Author

Darío Clavijo, Jun 22 2024

Status

approved