cp's OEIS Frontend

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

%I A375109 #50 Aug 19 2024 13:16:56
%S A375109 1,1,2,4,6,9,14,17,22,27,35,40,50,56,64,71,85,92,109,117,128,139,159,
%T A375109 168,182,194,208,219,245,256,285,298,314,331,349,361,396,414,433,448,
%U A375109 486,502,542,560,580,602,646,661,691,711,737,759,809
%N A375109 Number of distinct products i*j with 1 <= i, j <= n which are not the sum of two numbers between 1 and n.
%C A375109 In other words, these are the products that are not in {2..2*n}.
%C A375109 Essentialy each unique product i*j that is not i+j for 1 <= i, j <= n is in A254671+1.
%C A375109 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.
%C A375109 a(n) < A263995(n).
%F A375109 a(n) = A373716(n)+A108954(n).
%e A375109 a(3) = 2 because:
%e A375109  Prods = [1, 2, 3, 2, 4, 6, 3, 6, 9]
%e A375109  Sums = [2, 3, 4, 3, 4, 5, 4, 5, 6]
%e A375109  Items in Prods not in Sums : [1,9]
%e A375109 Total: 2.
%e A375109 a(4) = 4 because:
%e A375109  Prods = [1, 2, 3, 4, 2, 4, 6, 8, 3, 6, 9, 12, 4, 8, 12, 16]
%e A375109  Sums = [2, 3, 4, 5, 3, 4, 5, 6, 4, 5, 6, 7, 5, 6, 7, 8]
%e A375109  Items in Prods not in Sums: [1, 9, 12, 16]
%e A375109 Total: 4.
%o A375109 (Python)
%o A375109 def a(n):
%o A375109     P = {i * j for i in range(1, n+1) for j in range(1, n+1)}
%o A375109     return sum(1 for x in P if x > 2*n or x < 2)
%o A375109 print([a(n) for n in range(1,54)])
%o A375109 (Python)
%o A375109 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
%o A375109 (PARI) a(n) = #select(x->((x>2*n) || (x<2)), setbinop((x,y)->x*y, [1..n])); \\ _Michel Marcus_, Jul 30 2024
%Y A375109 Cf. A027424, A060715, A108954, A254671, A263995, A373716.
%K A375109 nonn,easy
%O A375109 1,3
%A A375109 _Darío Clavijo_, Jul 30 2024