A091626 -id:A091626 - cp's OEIS frontend

A094820 Partial sums of A038548.

1, 2, 3, 5, 6, 8, 9, 11, 13, 15, 16, 19, 20, 22, 24, 27, 28, 31, 32, 35, 37, 39, 40, 44, 46, 48, 50, 53, 54, 58, 59, 62, 64, 66, 68, 73, 74, 76, 78, 82, 83, 87, 88, 91, 94, 96, 97, 102, 104, 107, 109, 112, 113, 117, 119, 123, 125, 127, 128, 134, 135, 137, 140, 144, 146, 150

Offset: 1

Vladeta Jovovic, Jun 12 2004

a(n) = number of pairs (c,d) of integers such that 0 < c <= d and c*d <= n. - Clark Kimberling, Jun 18 2011

Equivalently, the number of representations of n in the form x + y*z, where x, y, and z are positive integers and y <= z. - John W. Layman, Feb 21 2012

Peter Kagey, Table of n, a(n) for n = 1..10000
Vaclav Kotesovec, Graph - the asymptotic ratio (100000 terms)

Cf. A091626, A038548.

Cf. A006218, A000196, A211264.

ListTools:-PartialSums([seq(ceil(numtheory:-tau(n)/2), n=1..100)]); # Robert Israel, Feb 24 2016

f[n_, k_] := Floor[n/k] - Floor[(n - 1)/k]
g[n_, k_] := If[k^2 <= n, f[n, k], 0]
Table[Sum[f[n, k], {k, 1, n}], {n, 1, 100}] (* A000005 *)
t = Table[Sum[g[n, k], {k, 1, n}], {n, 1, 100}]
(* A038548 *)
a[n_] := Sum[t[[i]], {i, 1, n}]
Table[a[n], {n, 1, 100}]  (* A094820 *)
(* Clark Kimberling, Jun 18 2011 *)
Table[Sum[Boole[d <= Sqrt[n]], {d, Divisors[n]}], {n, 1, 66}] // Accumulate (* Jean-François Alcover, Dec 13 2012 *)

a(n) = sum(k=1, n, ceil(numdiv(k)/2)); \\ Michel Marcus, Feb 24 2016

from math import isqrt
def A094820(n): return ((s:=isqrt(n))*(1-s)>>1)+sum(n//k for k in range(1,s+1)) # Chai Wah Wu, Oct 23 2023

def a(n)
    (1..Math.sqrt(n)).inject(0) { |accum, i| accum + 1 + (n/i).to_i - i }
  end # Peter Kagey, Feb 24 2016

G.f.: (1/(1 - x))*Sum_{k>=1} x^(k^2)/(1 - x^k). - Ilya Gutkovskiy, Apr 13 2017
a(n) ~ (log(n) + 2*gamma - 1)*n/2 + sqrt(n)/2, where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Aug 19 2019
a(n) = (A006218(n) + A000196(n))/2. - Ridouane Oudra, Nov 25 2022
a(n) = A211264(n) + A000196(n). - Ridouane Oudra, Sep 13 2024

A091627 Number of ordered integer pairs (b,c) with 1 <= b,c <= n such that both roots of x^2+bx+c=0 are integers.

Original entry on oeis.org

0, 0, 1, 2, 4, 5, 7, 8, 10, 12, 14, 15, 18, 19, 21, 23, 26, 27, 30, 31, 34, 36, 38, 39, 43, 45, 47, 49, 52, 53, 57, 58, 61, 63, 65, 67, 72, 73, 75, 77, 81, 82, 86, 87, 90, 93, 95, 96, 101, 103, 106, 108, 111, 112, 116, 118, 122, 124, 126, 127, 133, 134, 136, 139, 143

Offset: 0

Eric W. Weisstein, Jan 24 2004

nonn

Also number of ordered pairs of positive integers (i, j) such that i+j <= n and i*j <= n. - Seiichi Manyama, Sep 04 2021

Seiichi Manyama, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, Quadratic Equation

Cf. A000005, A067274, A091626, A094820.

Accumulate[ Join[{0, 0}, Table[ Ceiling[ DivisorSigma[0, n]/2], {n, 2, 64}]]]  (* Jean-François Alcover, Oct 23 2012, after Vladeta Jovovic *)

a(n) = sum(i=1, n, sum(j=i, n-i, i*j<=n)); \\ Seiichi Manyama, Sep 04 2021

N=66; x='x+O('x^N); concat([0, 0], Vec((-x+sum(k=1, sqrtint(N), x^k^2/(1-x^k)))/(1-x))) \\ Seiichi Manyama, Sep 04 2021

from math import isqrt
def A091627(n):
    m = isqrt(n)
    return 0 if n == 0 else sum(n//k for k in range(1, m+1))-m*(m-1)//2-1 # Chai Wah Wu, Oct 07 2021

a(n) = A091626(n) - n - 1. a(n) = a(n-1) + ceiling(tau(n)/2), n>1. Partial sums of A038548. - Vladeta Jovovic, Jun 12 2004

G.f.: (1/(1 - x)) * (-x + Sum_{k>=1} x^(k^2)/(1 - x^k)). - Seiichi Manyama, Sep 04 2021

A379597 a(n) is the number of distinct solution sets to the quadratic equations ux^2 + vx + w = 0 with integer coefficients u, v, w, abs(u) + abs(v) + abs(w) <= n having a nonnegative discriminant.

Original entry on oeis.org

1, 4, 12, 24, 50, 80, 134, 192, 276, 366, 510, 632, 834, 1018, 1262, 1502, 1858, 2136, 2584, 2956, 3448, 3910, 4576, 5076, 5834, 6488, 7320, 8066, 9136, 9892, 11118, 12114, 13358, 14482, 15978, 17108, 18862, 20272, 22024, 23532, 25700, 27216, 29600, 31486, 33746

Offset: 1

Felix Huber, Feb 18 2025

nonn

Quadratic equations u*x^2 + v*x + w = 0 with real coefficients u, v, w and nonnegative discriminant v^2 - 4*u*w have two real solutions.

a(n) is even for n >= 2.

			a(3) = 12 because there are 12 equations with abs(u) + abs(v) + abs(w) <= 3 and distinct solution set having a nonnegative discriminant: (u, v, w) = (1, 0, 0), (1, -1, 0), (1, 1, 0), (1, 0, -1), (1, -1, -1), (1, 1, -1), (1, -2, 0), (1, 2, 0), (1, 0, -2), (2, -1, 0), (2, 1, 0), and (2, 0, -1). Multiplied equations like 2*(1, 0, 0) = (2, 0, 0) or (-1)*(1, -1, 0) = (-1, 1, 0) do not have a distinct solution set.

Felix Huber, Table of n, a(n) for n = 1..3800
Eric Weisstein's World of Mathematics, Quadratic Equation

Cf. A067274, A091626, A091627, A364384, A364385, A365876, A365877, A381710, A381711.

A379597:=proc(n)
   option remember;
   local a,u,v,w;
   if n=1 then
      1
   else
      a:=0;	
      for u to n-1 do
         for v from 0 to n-u do
            w:=n-u-v;
            if igcd(u,v,w)=1 then
               if v=0 then
                  a:=a+1
               elif w=0 or w>=v^2/(4*u) then
                  a:=a+2
               else
                  a:=a+4
               fi
            fi
         od
      od;
      a+procname(n-1)
   fi;
end proc;
seq(A379597(n),n=1..45);

A381710 a(n) is the number of distinct solution sets to the quadratic equations ux^2 + vx + w = 0 with integer coefficients u, v, w, abs(u) + abs(v) + abs(w) <= n having a negative discriminant.

Original entry on oeis.org

0, 1, 5, 11, 25, 39, 69, 99, 143, 189, 265, 327, 437, 529, 653, 777, 965, 1107, 1343, 1531, 1783, 2021, 2367, 2619, 3013, 3343, 3771, 4153, 4707, 5087, 5721, 6229, 6865, 7437, 8197, 8767, 9677, 10391, 11279, 12043, 13155, 13919, 15147, 16101, 17249, 18301, 19763

Offset: 1

Felix Huber, Mar 06 2025

nonn

Quadratic equations u*x^2 + v*x + w = 0 with real coefficients u, v, w and negative discriminant v^2 - 4*u*w have two complex solutions.

a(n) is odd for n >= 2.

			a(3) = 5 because there are 5 equations with abs(u) + abs(v) + abs(w) <= 3 and distinct solution set having a negative discriminant: (u, v, w) = (1, 0, 1), (1, -1, 1), (1, 1, 1), (1, 0, 2), (2, 0, 1). Multiplied equations like (-1)*(1, -1, 1) = (-1, 1, -1) do not have a distinct solution set.

Felix Huber, Table of n, a(n) for n = 1..5000
Eric Weisstein's World of Mathematics, Quadratic Equation

Cf. A067274, A091626, A091627, A364384, A364385, A365876, A365877, A379597, A381711.

A381710:=proc(n)
   option remember;
   local a,u,v,w;
      if n=1 then
      0
   else
      a:=0;
      for u to n-1 do
         for v from 0 to n-u do
            w:=n-u-v;
               if igcd(u,v,w)=1 then
                  if v=0 then
                     a:=a+1
                  elif w>v^2/(4*u) then
                    a:=a+2
               fi
            fi
         od
      od;
      a+procname(n-1)
   fi;
end proc;
seq(A381710(n),n=1..47);

A381711 a(n) = A379597(n) - A381710(n).

Original entry on oeis.org

1, 3, 7, 13, 25, 41, 65, 93, 133, 177, 245, 305, 397, 489, 609, 725, 893, 1029, 1241, 1425, 1665, 1889, 2209, 2457, 2821, 3145, 3549, 3913, 4429, 4805, 5397, 5885, 6493, 7045, 7781, 8341, 9185, 9881, 10745, 11489, 12545, 13297, 14453, 15385, 16497, 17517, 18917

Offset: 1

Felix Huber, Mar 08 2025

nonn

a(n) is odd.

Felix Huber, Table of n, a(n) for n = 1..4051
Eric Weisstein's World of Mathematics, Quadratic Equation

Cf. A067274, A091626, A091627, A364384, A364385, A365876, A365877, A379597, A381710.

A381711:=proc(n)
   option remember;
   local a,u,v,w;
   if n=1 then
      1
   else
      a:=0;
      for u to n-1 do
         for v from 0 to n-u do
            w:=n-u-v;
            if igcd(u,v,w)=1 and v<>0 then
               if w=0 or w=v^2/(4*u) then
                  a:=a+2
               elif wA381711(n),n=1..47);

a(n) = A379597(n) - A381710(n).

A347275 a(n) is the number of nonnegative ordered pairs (a,b) satisfying (a+b <= n) and (a*b <= n).

Original entry on oeis.org

1, 3, 6, 10, 15, 19, 25, 29, 35, 40, 46, 50, 58, 62, 68, 74, 81, 85, 93, 97, 105, 111, 117, 121, 131, 136, 142, 148, 156, 160, 170, 174, 182, 188, 194, 200, 211, 215, 221, 227, 237, 241, 251, 255, 263, 271, 277, 281, 293, 298, 306, 312, 320, 324, 334, 340, 350, 356

Offset: 0

Vo Hoang Anh, Aug 25 2021

Except for a(0) and a(1), a(n) = A006218(n + 1) - A049820(n + 1) + 2n - 1. Tested with first 10^9 numbers.
Since there are only O(sqrt(n)) different integers in the set S = {floor(n / 1), floor(n / 2), ..., floor(n / n)}, we can calculate a(n) in O(sqrt(n)).
There are many methods based on different kinds of formulas that are also known to work in O(n^(1/3)), but not all are effective. Some of them have lower complexity, but their use of too much division and multiplication makes them slower than expected; some of them use precalculation and caching to speed things up but are not as effective as the division-free algorithm that Richard Sladkey described.
a(n) is approximately epsilon*n^(1+delta) as delta approaches 0 and epsilon approaches +oo for n approaching +oo. The tighter relationship between epsilon and delta is unknown. Tested with 10^7 numbers n <= 10^15 using power regression algorithm.
For n > 1, a(n) = A006218(n)+2n-1. - Chai Wah Wu, Oct 07 2021

			a(1) = 3: (0, 0), (0, 1), (1, 0).

Richard Sladkey, A Successive Approximation Algorithm for Computing the Divisor Summatory Function, arXiv:1206.3369 [math.NT], 2012.

Cf. A006218, A049820, A091626.

int T(int n) { int cnt = 0; for (int a = 0; a <= n; ++a) for (int b = 0; b <= n - a; ++b) if (a * b <= n) ++cnt; return cnt; }

A347275 := proc(n)
    local a,i,j ;
    a := 0 ;
    for i from 0 to n do
    for j from 0 to n-i do
        if i*j <= n then
            a := a+1 ;
        end if;
    end do:
    end do:
    a ;
end proc:
seq(A347275(n),n=0..40) ; # R. J. Mathar, Sep 15 2021

a[n_] := Sum[Boole[i*j <= n], {i, 0, n}, {j, 0, n-i}];
Table[a[n], {n, 0, 60}] (* Jean-François Alcover, Nov 08 2023 *)

from math import isqrt
def A347275(n): return 2*n+1 if n <= 1 else 2*(n+sum(n//k for k in range(1, isqrt(n)+1)))-isqrt(n)**2-1 # Chai Wah Wu, Oct 07 2021

a(n) = Sum_{a=0..n} Sum_{b=0..n-a} [a*b<=n], where [] is the Iverson bracket.

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A094820 Partial sums of A038548.

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A091627 Number of ordered integer pairs (b,c) with 1 <= b,c <= n such that both roots of x^2+bx+c=0 are integers.

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A379597 a(n) is the number of distinct solution sets to the quadratic equations u*x^2 + v*x + w = 0 with integer coefficients u, v, w, abs(u) + abs(v) + abs(w) <= n having a nonnegative discriminant.

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A381710 a(n) is the number of distinct solution sets to the quadratic equations u*x^2 + v*x + w = 0 with integer coefficients u, v, w, abs(u) + abs(v) + abs(w) <= n having a negative discriminant.

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A381711 a(n) = A379597(n) - A381710(n).

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A347275 a(n) is the number of nonnegative ordered pairs (a,b) satisfying (a+b <= n) and (a*b <= n).

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A379597 a(n) is the number of distinct solution sets to the quadratic equations ux^2 + vx + w = 0 with integer coefficients u, v, w, abs(u) + abs(v) + abs(w) <= n having a nonnegative discriminant.

A381710 a(n) is the number of distinct solution sets to the quadratic equations ux^2 + vx + w = 0 with integer coefficients u, v, w, abs(u) + abs(v) + abs(w) <= n having a negative discriminant.