A108407 - OEIS

%I #18 Oct 13 2023 17:49:04

%S 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,

%T 17,15,16,15,23,0,18,18,24,0,25,0,24,24,22,0,31,18,28,24,29,0,32,24,

%U 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

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

%H Seiichi Manyama, <a href="/A108407/b108407.txt">Table of n, a(n) for n = 1..1000</a>

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

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

%e When going to 8 X 8, the added entries 8,16,24 are already known, so a(7)=3:

%e .1..2..3..4..5..6..7....8 *

%e ....4..6..8.10.12.14...16 *

%e .......9.12.15.18.21...24 *

%e .........16.20.24.28...32

%e ............25.30.35...40

%e ...............36.42...48

%e ..................49...56

%e .......................64

%p A108407 := proc(n)

%p     n+1-A062854(n+1) ;

%p end proc:

%p seq(A108407(n),n=1..40) ; # _R. J. Mathar_, Oct 02 2020

%t nmax = 100;

%t A062854 = Table[u = If[n == 1, {}, Union[u, n Range[n]]]; Length[u], {n, 1, nmax+1}] // Differences // Prepend[#, 1]&;

%t a[n_] := n + 1 - A062854[[n+1]];

%t Table[a[n], {n, 1, nmax}] (* _Jean-François Alcover_, Oct 02 2020 *)

%o (Python)

%o from itertools import takewhile

%o from python import divisors

%o 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

%Y Unique values of sequence are in A108408.

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

%K nonn

%O 1,5

%A _Ralf Stephan_, Jun 03 2005