cp's OEIS Frontend

A106633 Number of ways to express n as k+l*m, with k, l, m all in the range [0..n].

%I A106633 #35 May 12 2024 11:35:02
%S A106633 1,4,8,12,17,21,27,31,37,42,48,52,60,64,70,76,83,87,95,99,107,113,119,
%T A106633 123,133,138,144,150,158,162,172,176,184,190,196,202,213,217,223,229,
%U A106633 239,243,253,257,265,273,279,283,295,300,308,314,322,326,336,342,352
%N A106633 Number of ways to express n as k+l*m, with k, l, m all in the range [0..n].
%C A106633 Number of ordered triples [k,l,m] with n = k+l*m and k, l, m all in the range [0..n].
%C A106633 From _R. J. Mathar_, Jun 30 2013: (Start)
%C A106633 A010766 is the following array A read by antidiagonals:
%C A106633   1,  1,  1,  1,  1,  1, ...
%C A106633   2,  1,  1,  1,  1,  1, ...
%C A106633   3,  2,  1,  1,  1,  1, ...
%C A106633   4,  2,  2,  1,  1,  1, ...
%C A106633   5,  3,  2,  2,  1,  1, ...
%C A106633   6,  3,  2,  2,  2,  1, ...
%C A106633 and apparently a(n) is the hook sum Sum_{k=0..n} A(n,k) + Sum_{r=0..n-1} A(r,n). (End)
%H A106633 R. J. Mathar, <a href="/A106633/b106633.txt">Table of n, a(n) for n = 0..1000</a>
%F A106633 From _Ridouane Oudra_, Apr 22 2024: (Start)
%F A106633 a(n) = 2*n + 1 + Sum_{k=1..n} floor(n/k);
%F A106633 a(n) = 2*n + 1 + Sum_{k=1..n} tau(k);
%F A106633 a(n) = A005408(n) + A006218(n). (End)
%e A106633 0+1*2 = 0+2*1 = 1+1*1 = 2+0*0 = 2+0*1 = 2+0*2 = 2+1*0 = 2+2*0 = 2, so a(2)=8.
%e A106633 a(3)=12: 3+0*0, 3+0*m (6), 2+1*1, 1+2*1 (2), 0+3*1 (2).
%p A106633 A106633 := proc(n)
%p A106633     local a, k, l, m ;
%p A106633     a := 0 ;
%p A106633     for k from 0 to n do
%p A106633         for l from 0 to n do
%p A106633             if l = 0 then
%p A106633                 if k = n then
%p A106633                     a := a+n+1 ;
%p A106633                 end if;
%p A106633             else
%p A106633                 m := (n-k)/l ;
%p A106633                 if type(m,'integer') then
%p A106633                     a := a+1 ;
%p A106633                 end if;
%p A106633             end if;
%p A106633         end do:
%p A106633     end do:
%p A106633     a ;
%p A106633 end proc: # _R. J. Mathar_, Oct 17 2012
%t A106633 A106633[n_] := Module[{a, m}, a = 0; Do[If[l == 0, If[k == n, a = a + n + 1], m = (n - k)/l; If[IntegerQ[m], a = a + 1]], {k, 0, n}, {l, 0, n}]; a];
%t A106633 Table[A106633[n], {n, 0, 56}] (* _Jean-François Alcover_, Jun 10 2023, after _R. J. Mathar_ *)
%o A106633 (PARI) list(n)={
%o A106633     my(v=vector(n),t);
%o A106633     for(i=2,n,for(j=1,min(n\i,i-1),v[i*j]+=2));
%o A106633     for(i=1,sqrtint(n),v[i^2]++);
%o A106633     concat(1,vector(n,k,2*k+1+t+=v[k]))
%o A106633 }; \\ _Charles R Greathouse IV_, Oct 17 2012
%Y A106633 Cf. A006218, A106634, A106846, A106847.
%Y A106633 Cf. A005408, A006218.
%K A106633 nonn,easy
%O A106633 0,2
%A A106633 _Ralf Stephan_, May 06 2005
%E A106633 Definition clarified by _N. J. A. Sloane_, Jul 07 2012