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

%I A108407 #21 Apr 28 2025 12:20:59
%S A108407 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 A108407 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 A108407 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 A108407 Number of added unique known entries when going from the n X n to the (n+1) X (n+1) multiplication table.
%H A108407 Seiichi Manyama, <a href="/A108407/b108407.txt">Table of n, a(n) for n = 1..1000</a>
%F A108407 For prime p, a(p-1) = 0.
%F A108407 a(n) = n+1 - A062854(n+1).
%e A108407 When going to 8 X 8, the added entries 8,16,24 are already known, so a(7)=3:
%e A108407 .1..2..3..4..5..6..7....8 *
%e A108407 ....4..6..8.10.12.14...16 *
%e A108407 .......9.12.15.18.21...24 *
%e A108407 .........16.20.24.28...32
%e A108407 ............25.30.35...40
%e A108407 ...............36.42...48
%e A108407 ..................49...56
%e A108407 .......................64
%p A108407 A108407 := proc(n)
%p A108407     n+1-A062854(n+1) ;
%p A108407 end proc:
%p A108407 seq(A108407(n),n=1..40) ; # _R. J. Mathar_, Oct 02 2020
%t A108407 nmax = 100;
%t A108407 A062854 = Table[u = If[n == 1, {}, Union[u, n Range[n]]]; Length[u], {n, 1, nmax+1}] // Differences // Prepend[#, 1]&;
%t A108407 a[n_] := n + 1 - A062854[[n+1]];
%t A108407 Table[a[n], {n, 1, nmax}] (* _Jean-François Alcover_, Oct 02 2020 *)
%o A108407 (Python)
%o A108407 from itertools import takewhile
%o A108407 from sympy import divisors
%o A108407 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 A108407 Unique values of sequence are in A108408.
%Y A108407 Cf. A027424 (total unique entries), A062854 (added unique unknown entries).
%K A108407 nonn
%O A108407 1,5
%A A108407 _Ralf Stephan_, Jun 03 2005