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

0, 1, 3, 7, 11, 17, 23, 31, 40, 50, 60, 72, 84, 98, 113, 129, 145, 163, 181, 201, 222, 244, 266, 290, 315, 341, 368, 396, 424, 454, 484, 516, 549, 583, 618, 654, 690, 728, 767, 807, 847, 889, 931, 975, 1020, 1066, 1112, 1160, 1209, 1259, 1310, 1362, 1414

Offset

1,3

Comments

The number of pairs (i,j) with 1 <= i <= j <= n and i*j composite. - Peter Kagey, Sep 24 2020

Links

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

Formula

a(n) = A000217(n) - A000720(n) - 1. - David A. Corneth, Sep 08 2020

a(n) = A256885(n) - 1. - Michel Marcus, Sep 09 2020

a(n+1) - a(n) = A014684(n+1). - Bill McEachen, Oct 30 2020

Example

There are a(7) = 23 composite numbers in the 7x7 triangular multiplication table with the hypotenuse being the Square numbers:
1   2   3   4*  5   6*  7
    4*  6*  8* 10* 12* 14*
        9* 12* 15* 18* 21*
           16* 20* 24* 28*
               25* 30* 35*
                   36* 42*
                       49*

Mathematica

Array[Binomial[# + 1, 2] - PrimePi[#] - 1 &, 53] (* Michael De Vlieger, Nov 05 2020 *)

Prog

(PARI) a(n) = binomial(n+1, 2) - primepi(n)-1 \\ David A. Corneth, Sep 08 2020
(Python)
from sympy import primepi
def A333996(n): return (n*(n+1)>>1)-primepi(n)-1 # Chai Wah Wu, Oct 14 2023

Crossrefs

Cf. A014684 (first differences), A333995, A108407, A000720, A000217, A256885, A334454.

Sequence in context: A134707 A047838 A188653 * A262502 A211076 A180452

Adjacent sequences: A333993 A333994 A333995 * A333997 A333998 A333999

Keyword

nonn,easy

Author

Charles Kusniec, Sep 05 2020

Status

approved