A333995 a(n) = number of distinct composite numbers in the n X n multiplication table that are not in the n-1 X n-1 multiplication table.

0, 1, 2, 3, 4, 4, 6, 5, 6, 6, 10, 6, 12, 8, 9, 8, 16, 9, 18, 10, 12, 12, 22, 10, 16, 14, 15, 13, 28, 12, 30, 15, 18, 18, 20, 13, 36, 20, 21, 16, 40, 17, 42, 20, 21, 24, 46, 17, 31, 22, 27, 23, 52, 22, 31, 22, 30, 30, 58, 19, 60, 32, 28, 26, 36, 26, 66, 30, 36

Offset

1,3

Comments

From Charles Kusniec, Jun 23 2026: (Start)

Equivalently, a(n) counts the composite products n*k, 1 <= k <= n, such that A033677(n*k) = n. These are the composite primitive entries of column n in the triangular multiplication table.

For prime p, a(p) = p-1, since all products p*k, 1 <= k <= p, first appear in column p; the term p itself is prime and therefore is not counted by a(n).

Apart from the initial exceptional record at n = 4, the record values of a(n) occur at n = prime.

For n > 1, a(n) + A010051(n) is the length of row n of A397409. Because a(p) + A010051(p) = p, every prime p is a record position of this row-length sequence. (End)

Links

Table of n, a(n) for n=1..69.

Formula

a(n) = n - A108407(n-1) - A010051(n), n > 1. - Corrected by R. J. Mathar, Oct 02 2020

a(n) = A062854(n) - A010051(n) for n > 1. - Chai Wah Wu, Oct 14 2023

Example

a(2) = 1 since the 1 X 1 and 2 X 2 multiplication tables are
---
  1
---
  2 4
  1 2
---
and the composite number 4 has appeared.
---
a(8)=5:
                                  64*
                             49   56*
                        36   42   48*
                   25   30   35   40*
              16   20   24   28   32*
          9   12   15   18   21   24
     4    6    8   10   12   14   16
1    2    3    4    5    6    7    8

Mathematica

a[n_] := Sum[Boole[AllTrue[Select[Divisors[n*k], # < n &], # <= k &]], {k, 1, n}] - Boole[PrimeQ[n]]; a[1] = 0; Array[a, 70] (* Amiram Eldar, Jun 27 2026 *)

Prog

(Python)
from itertools import takewhile
from sympy import divisors, isprime
def A333995(n): return sum(1 for i in range(1, n+1) if all(d<=i for d in takewhile(lambda d:d<n, divisors(n*i))))-int(isprime(n)) if n>1 else 0 # Chai Wah Wu, Oct 14 2023

Crossrefs

Cf. A010051, A062854, A108407, A033677.

Sequence in context: A298933 A365851 A091860 * A397104 A301764 A181833

Adjacent sequences: A333992 A333993 A333994 * A333996 A333997 A333998

Keyword

nonn,changed

Author

Charles Kusniec, Sep 05 2020

Status

approved