A108407 Number of added unique known entries when going from the n X n to the (n+1) X (n+1) multiplication table.

0, 0, 1, 0, 2, 0, 3, 3, 4, 0, 6, 0, 6, 6, 8, 0, 9, 0, 10, 9, 10, 0, 14, 9, 12, 12, 15, 0, 18, 0, 17, 15, 16, 15, 23, 0, 18, 18, 24, 0, 25, 0, 24, 24, 22, 0, 31, 18, 28, 24, 29, 0, 32, 24, 34, 27, 28, 0, 41, 0, 30, 35, 38, 29, 40, 0, 38, 33, 44, 0, 49, 0, 36, 41, 43, 32, 47, 0, 52

Offset: 1

Author

Ralf Stephan, Jun 03 2005

Keywords

• nonn

Examples

			When going to 8 X 8, the added entries 8,16,24 are already known, so a(7)=3:
.1..2..3..4..5..6..7....8 *
....4..6..8.10.12.14...16 *
.......9.12.15.18.21...24 *
.........16.20.24.28...32
............25.30.35...40
...............36.42...48
..................49...56
.......................64
		

Links

• Seiichi Manyama, Table of n, a(n) for n = 1..1000

Crossrefs

Unique values of sequence are in A108408.

Cf. A027424 (total unique entries), A062854 (added unique unknown entries).

Programs

Maple

A108407 := proc(n)
    n+1-A062854(n+1) ;
end proc:
seq(A108407(n),n=1..40) ; # R. J. Mathar, Oct 02 2020

Mathematica

nmax = 100;
A062854 = Table[u = If[n == 1, {}, Union[u, n Range[n]]]; Length[u], {n, 1, nmax+1}] // Differences // Prepend[#, 1]&;
a[n_] := n + 1 - A062854[[n+1]];
Table[a[n], {n, 1, nmax}] (* Jean-François Alcover, Oct 02 2020 *)

Python

from itertools import takewhile
from sympy import divisors
def A108407(n): return n+1-sum(1 for i in range(1,n+2) if all(d<=i for d in takewhile(lambda d:d<=n,divisors((n+1)*i)))) # Chai Wah Wu, Oct 13 2023

Formula

For prime p, a(p-1) = 0.

a(n) = n+1 - A062854(n+1).