A375109 Number of distinct products i*j with 1 <= i, j <= n which are not the sum of two numbers between 1 and n.

1, 1, 2, 4, 6, 9, 14, 17, 22, 27, 35, 40, 50, 56, 64, 71, 85, 92, 109, 117, 128, 139, 159, 168, 182, 194, 208, 219, 245, 256, 285, 298, 314, 331, 349, 361, 396, 414, 433, 448, 486, 502, 542, 560, 580, 602, 646, 661, 691, 711, 737, 759, 809

Offset

1,3

Comments

In other words, these are the products that are not in {2..2*n}.

Essentialy each unique product i*j that is not i+j for 1 <= i, j <= n is in A254671+1.

Conversely the number of distinct sums i+j with 1 <= i, j <= n which are not the product of two numbers between 1 and n is A060715.

a(n) < A263995(n).

Links

Table of n, a(n) for n=1..53.

Formula

a(n) = A373716(n)+A108954(n).

Example

a(3) = 2 because:
 Prods = [1, 2, 3, 2, 4, 6, 3, 6, 9]
 Sums = [2, 3, 4, 3, 4, 5, 4, 5, 6]
 Items in Prods not in Sums : [1,9]
Total: 2.
a(4) = 4 because:
 Prods = [1, 2, 3, 4, 2, 4, 6, 8, 3, 6, 9, 12, 4, 8, 12, 16]
 Sums = [2, 3, 4, 5, 3, 4, 5, 6, 4, 5, 6, 7, 5, 6, 7, 8]
 Items in Prods not in Sums: [1, 9, 12, 16]
Total: 4.

Prog

(Python)
def a(n):
    P = {i * j for i in range(1, n+1) for j in range(1, n+1)}
    return sum(1 for x in P if x > 2*n or x < 2)
print([a(n) for n in range(1, 54)])
(Python)
def A375109(n): return len({i*j for i in range(1, n+1) for j in range((n<<1)//i+1, i+1)})+1 # Chai Wah Wu, Aug 19 2024
(PARI) a(n) = #select(x->((x>2*n) || (x<2)), setbinop((x, y)->x*y, [1..n])); \\ Michel Marcus, Jul 30 2024

Crossrefs

Cf. A027424, A060715, A108954, A254671, A263995, A373716.

Sequence in context: A164140 A205883 A253108 * A198201 A281989 A328423

Adjacent sequences: A375106 A375107 A375108 * A375110 A375111 A375112

Keyword

nonn,easy

Author

Darío Clavijo, Jul 30 2024

Status

approved