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A281376 Total number of counts where floor(N/k) < floor((N+k)/n) for k = {1, 2, ..., n-1} and N >= n. 1
0, 0, 0, 1, 3, 6, 11, 17, 25, 35, 47, 60, 77, 95, 115, 138, 164, 191, 222, 254, 290, 329, 370, 412, 460, 510, 562, 617, 676, 736, 802, 869, 940, 1014, 1090, 1169, 1255, 1342, 1431, 1523, 1621, 1720, 1825, 1931, 2041, 2156, 2273, 2391, 2517, 2645, 2777 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,5

LINKS

Jon E. Schoenfield, Table of n, a(n) for n = 1..10000 (terms 1..200 from Lorenz H. Menke, Jr.)

FORMULA

a(n) = Sum_{d=1..ceiling((n-3)/3)} Sum_{j=1..n-(2*d+1)} floor(j/d). - Jon E. Schoenfield, Jan 23 2017

a(n) = Sum_{d=1..ceiling(n/3)-1} ((j+1)*(j*d/2 + n mod d)), where j = floor(n/d) - 3. - Jon E. Schoenfield, Jan 24 2017

EXAMPLE

For n = 5, we have counted the cases where floor(N/k) < floor((N+k)/5), k = {1,2,3,4} then a(5) = 3, since this is true only for k = 4 and N = 6, k = 4 and N = 7, and k = 4 and N = 11.

MAPLE

A281376 := proc(n)

    local a, k, N;

    a := 0 ;

    for k from 1 to n-1 do

        for N from n do

            if floor(N/k) < floor((N+k)/n) then

                a := a +1 ;

            elif N >= (k+2*n)*k/(n-k) then

                break;

            end if;

        end do:

    end do:

    a ;

end proc:

seq(A281376(n), n=1..10) ; # R. J. Mathar, Feb 03 2017

MATHEMATICA

a[n_] :=

  Block[{lhs, rhs, count},

   count = 0;

   Do[lhs = Floor[H1/k];

    rhs = Floor[(H1 + k)/n];

    If[lhs < rhs, count++], {k, 1, n - 1}, {H1,

     n, (n^2 - 3 n + 1) + 10}]; (* the 10 is simply guard counts *)

   Return[count]];

a281376[n_] :=

Sum[Floor[j/d], {d, Ceiling[(n - 3)/3]}, {j, n - (2 d + 1)}]

(* Hartmut F. W. Hoft, Jan 25 2017; based on the first formula above *)

PROG

(PARI) a(n) = sum(d = 1, ceil((n-3)/3), sum(j = 1, n-(2*d+1), j\d)); \\ Michel Marcus, Jan 29 2017

CROSSREFS

Sequence in context: A273140 A320272 A119639 * A247586 A107957 A000603

Adjacent sequences:  A281373 A281374 A281375 * A281377 A281378 A281379

KEYWORD

nonn

AUTHOR

Lorenz H. Menke, Jr., Jan 20 2017

STATUS

approved

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Last modified January 25 08:59 EST 2021. Contains 340416 sequences. (Running on oeis4.)