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A333996 Number of composite numbers in the triangular n X n multiplication table.

%I A333996 #46 Oct 14 2023 11:34:05
%S A333996 0,1,3,7,11,17,23,31,40,50,60,72,84,98,113,129,145,163,181,201,222,
%T A333996 244,266,290,315,341,368,396,424,454,484,516,549,583,618,654,690,728,
%U A333996 767,807,847,889,931,975,1020,1066,1112,1160,1209,1259,1310,1362,1414
%N A333996 Number of composite numbers in the triangular n X n multiplication table.
%C A333996 The number of pairs (i,j) with 1 <= i <= j <= n and i*j composite. - _Peter Kagey_, Sep 24 2020
%F A333996 a(n) = A000217(n) - A000720(n) - 1. - _David A. Corneth_, Sep 08 2020
%F A333996 a(n) = A256885(n) - 1. - _Michel Marcus_, Sep 09 2020
%F A333996 a(n+1) - a(n) = A014684(n+1). - _Bill McEachen_, Oct 30 2020
%e A333996 There are a(7) = 23 composite numbers in the 7x7 triangular multiplication table with the hypotenuse being the Square numbers:
%e A333996 1   2   3   4*  5   6*  7
%e A333996     4*  6*  8* 10* 12* 14*
%e A333996         9* 12* 15* 18* 21*
%e A333996            16* 20* 24* 28*
%e A333996                25* 30* 35*
%e A333996                    36* 42*
%e A333996                        49*
%t A333996 Array[Binomial[# + 1, 2] - PrimePi[#] - 1 &, 53] (* _Michael De Vlieger_, Nov 05 2020 *)
%o A333996 (PARI) a(n) = binomial(n+1, 2) - primepi(n)-1 \\ _David A. Corneth_, Sep 08 2020
%o A333996 (Python)
%o A333996 from sympy import primepi
%o A333996 def A333996(n): return (n*(n+1)>>1)-primepi(n)-1 # _Chai Wah Wu_, Oct 14 2023
%Y A333996 Cf. A014684 (first differences), A333995, A108407, A000720, A000217, A256885, A334454.
%K A333996 nonn,easy
%O A333996 1,3
%A A333996 _Charles Kusniec_, Sep 05 2020