A094820 -id:A094820 - cp's OEIS frontend

A211275 Number of integer pairs (x,y) such that 0 < x <= y <= n and x*y <= floor(n/2).

0, 1, 1, 2, 2, 3, 3, 5, 5, 6, 6, 8, 8, 9, 9, 11, 11, 13, 13, 15, 15, 16, 16, 19, 19, 20, 20, 22, 22, 24, 24, 27, 27, 28, 28, 31, 31, 32, 32, 35, 35, 37, 37, 39, 39, 40, 40, 44, 44, 46, 46, 48, 48, 50, 50, 53, 53, 54, 54, 58, 58, 59, 59, 62, 62, 64, 64, 66, 66, 68, 68

Offset: 1

Clark Kimberling, Apr 07 2012

nonn

For a guide to related sequences, see A211266.

Cf. A211266.

a = 1; b = n; z1 = 120;
t[n_] :=  t[n] = Flatten[Table[x*y, {x, a, b - 1},
{y, x, b}]]
c[n_, k_] := c[n, k] = Count[t[n], k]
Table[c[n, n], {n, 1, z1}]           (* A038548 *)
Table[c[n, n + 1], {n, 1, z1}]       (* A072670 *)
Table[c[n, 2*n], {n, 1, z1}]         (* A211270 *)
Table[c[n, 3*n], {n, 1, z1}]         (* A211271 *)
Table[c[n, Floor[n/2]], {n, 1, z1}]  (* A211272 *)
c1[n_, m_] := c1[n, m] = Sum[c[n, k], {k, a, m}]
Print
Table[c1[n, n], {n, 1, z1}]          (* A094820 *)
Table[c1[n, n + 1], {n, 1, z1}]      (* A091627 *)
Table[c1[n, 2*n], {n, 1, z1}]        (* A211273 *)
Table[c1[n, 3*n], {n, 1, z1}]        (* A211274 *)
Table[c1[n, Floor[n/2]], {n, 1, z1}] (* A211275 *)

A161842 Partial sums of A161841.

Original entry on oeis.org

2, 4, 6, 10, 12, 16, 18, 22, 26, 30, 32, 38, 40, 44, 48, 54, 56, 62, 64, 70, 74, 78, 80, 88, 92, 96, 100, 106, 108, 116, 118, 124, 128, 132, 136, 146, 148, 152, 156, 164, 166, 174, 176, 182, 188, 192, 194, 204, 208, 214, 218, 224, 226, 234, 238, 246

Offset: 1

Omar E. Pol, Jun 23 2009

David A. Corneth, Table of n, a(n) for n = 1..10000
Michael Penn, This is always even??, YouTube video, 2020.

Cf. A001620, A008836, A038548, A094820, A161841.

Accumulate[2*Ceiling[DivisorSigma[0, Range[100]]/2]] (* Paolo Xausa, Feb 05 2025 *)

a(n) = sum(i=1, n, floor(n/i)) + sqrtint(n) \\ David A. Corneth, Dec 17 2020

first(n) = {my(res = vector(n), t = 0); for(i = 1, n, t+=(numdiv(i)+issquare(i)); res[i] = t ); res } \\ David A. Corneth, Dec 17 2020

a(n) = 2*A094820(n).
a(n) = Sum_{i=1..n} (1 + A008836(i))*floor(n/i). - Enrique Pérez Herrero, Jul 10 2012
a(n) ~ (log(n) + 2*gamma - 1)*n + sqrt(n), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Jul 01 2021

A347517 Partial sums of A347516.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 9, 10, 12, 13, 15, 16, 18, 19, 21, 22, 24, 25, 27, 28, 30, 31, 33, 34, 36, 38, 40, 41, 44, 45, 47, 49, 51, 52, 55, 56, 58, 60, 62, 63, 66, 67, 69, 71, 73, 74, 77, 78, 80, 82, 84, 85, 88, 89, 91, 93, 95, 96, 99, 100, 102, 104, 107, 108, 111, 112, 115, 117, 119, 120, 124, 125, 127, 129, 132, 133, 136

Offset: 1

Seiichi Manyama, Sep 04 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A006218, A048766, A094820, A347516.

f[n_] := DivisorSum[n, 1 &, # <= n^(1/3) &]; Accumulate @ Array[f, 100] (* Amiram Eldar, Sep 04 2021 *)

N=99; x='x+O('x^N); Vec(sum(k=1, N^(1/3), x^k^3/(1-x^k))/(1-x))

a(n) = sum(k=1, n, sumdiv(k, d, d^3<=k)); \\ Michel Marcus, Sep 05 2021

G.f.: (1/(1 - x)) * Sum_{k>=1} x^(k^3)/(1 - x^k).

A347527 Partial sums of A347526.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 17, 18, 20, 21, 23, 24, 26, 27, 29, 30, 32, 33, 35, 36, 38, 39, 41, 42, 44, 45, 47, 48, 50, 51, 53, 54, 56, 57, 59, 60, 62, 63, 65, 66, 68, 69, 71, 72, 74, 75, 77, 78, 80, 81, 83, 84, 86, 87, 89, 90, 92, 93, 95, 96, 98, 99, 101, 102

Offset: 1

Seiichi Manyama, Sep 05 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A006218, A013938, A094820, A347517, A347526.

f[n_] := DivisorSum[n, 1 &, # <= n^(1/4) &]; Accumulate @ Array[f, 100] (* Amiram Eldar, Sep 05 2021 *)

a(n) = sum(k=1, n, sumdiv(k, d, d^4<=k));

N=99; x='x+O('x^N); Vec(sum(k=1, N^(1/4), x^k^4/(1-x^k))/(1-x))

G.f.: (1/(1 - x)) * Sum_{k>=1} x^(k^4)/(1 - x^k).

A212597 Number of ways of writing n in the form ij+km with 0

Original entry on oeis.org

0, 0, 1, 1, 3, 3, 5, 5, 7, 9, 10, 11, 14, 15, 16, 19, 20, 23, 24, 27, 28, 33, 30, 37, 36, 42, 40, 48, 44, 53, 49, 57, 55, 65, 55, 72, 64, 74, 70, 83, 72, 90, 77, 95, 87, 102, 84, 112, 94, 112, 104, 124, 102, 133, 109, 135, 123, 142, 117, 160, 128, 152, 138

Offset: 1

John W. Layman, May 22 2012

			1*1+1*4 = 1*2+1*3 = 1*1+2*2 = 5, so a(5) = 3.

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Cf. A007875, A094820.

with(numtheory):
a:= proc(n) local j, l, m;
      add(add(add(`if`(j is(h>=sqrt(n-l)), divisors(n-l))),
      j=select(h-> is(h>=sqrt(l)), divisors(l))), l=1..n-1)
    end:
seq(a(n), n=1..100);  # Alois P. Heinz, May 24 2012

a[n_] := Sum[Sum[Sum[If[j < m || j == m && l*m < (n-l)*j, 1, 0], {m, Select[Divisors[n-l], # >= Sqrt[n-l]&]}], {j, Select[Divisors[l], # >= Sqrt[l]&]}], {l, 1, n-1}];
Array[a, 100] (* Jean-François Alcover, Mar 27 2017, after Alois P. Heinz *)

A255315 Lower triangular matrix describing the shape of a half hyperbola in the Dirichlet divisor problem.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 0, 2, 1, 1, 0, 2, 1, 1, 1, 0, 1, 2, 1, 1, 1, 0, 1, 2, 1, 1, 1, 1, 0, 1, 1, 2, 1, 1, 1, 1, 0, 0, 2, 2, 1, 1, 1, 1, 1, 0, 0, 2, 1, 2, 1, 1, 1, 1, 1, 0, 0, 2, 1, 2, 1, 1, 1, 1, 1, 1, 0, 0, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1

Offset: 1

Mats Granvik, May 31 2015

The sum of terms of row n is n. Length of row n is n.
From Mats Granvik, Feb 21 2016: (Start)
A006218(n) = (n^2 - ((2*Sum_{kk=1..n} Sum_{k=1..kk} T(n,k)) - n)) + 2*n - round(1 + (1/2)*(-3 + sqrt(n) + sqrt(1 + n))).
A006218(n) = -((n^2 - ((2*Sum_{kk=1..n} Sum_{k=1..kk} T(n,n-k+1)) - n)) - 2*n + round(1 + (1/2)*(-3 + sqrt(n) + sqrt(1 + n)))).
(End)
From Mats Granvik, May 28 2017: (Start)
A006218(n) = (n^2 - (2*(Sum_{k=1..n} T(n, k)*(n - k + 1)) - n)) + 2*n - round(1 + (1/2)*(-3 + sqrt(n) + sqrt(1 + n))).
A006218(n) = -((n^2 - (2*(Sum_{k=1..n} T(n, n - k + 1)*(n - k + 1)) - n)) - 2*n + round(1 + (1/2)*(-3 + sqrt(n) + sqrt(1 + n)))).
(End)
From Mats Granvik, Sep 07 2017: (Start)
It appears that:
The number of 0's in row n is equal to the number of 2's in row n and their number is given by A000196(n) - 1.
The number of 1's in column k is given by A152948(k+2).
The number of 2's in column k is given by A000096(k-1).
The row index of the last nonzero entry in column k is given by A005563(k).
(End)
From Mats Granvik, Oct 06 2018: (Start)
The smallest k such that T(n,k)=2 is given by A079643(n) = floor(n/floor(sqrt(n))).
This gives the lower bound: A006218(n) >= A094761(n) + A079643(n)*2*(A000196(n)-1).
<=> A006218(n) >= 2*n - (floor(sqrt(n)))^2 + floor(n/floor(sqrt(n)))*2*floor(sqrt(n)-1).
The average of k:s such that T(n,k)=2, for n>3 is given by:
b(n) = Sum_{k=1..n} (k*floor(abs(T(n, k)-1/2)))/floor(sqrt(n)-1).
This gives A006218(n) = 2*n - (floor(sqrt(n)))^2 + b(n)*2*floor(sqrt(n)-1) = 2*n - (floor(sqrt(n)))^2 + (Sum_{k=1..n} (k*floor(abs(T(n, k)-1/2))))*2, for n>3.
The largest k such that T(n,k)=2 is given by A004526(n) = floor(n/2).
This gives the upper bound: A006218(n) <= A094761(n) + A004526(n)*2*(A000196(n)-1).
<=> A006218(n) <= 2*n - (floor(sqrt(n)))^2 + floor(n/2)*2*floor(sqrt(n)-1).
The lower bound starts:  1, 3, 5, 8, 10, 14, 16, 20, 21, 23, ...
Sequence A006218 starts: 1, 3, 5, 8, 10, 14, 16, 20, 23, 27, ...
The upper bound starts:  1, 3, 5, 8, 10, 14, 16, 20, 25, 31, ...
(End)

			1;
1, 1;
1, 1, 1;
0, 2, 1, 1;
0, 2, 1, 1, 1;
0, 1, 2, 1, 1, 1;
0, 1, 2, 1, 1, 1, 1;
0, 1, 1, 2, 1, 1, 1, 1;
0, 0, 2, 2, 1, 1, 1, 1, 1;
0, 0, 2, 1, 2, 1, 1, 1, 1, 1;
0, 0, 2, 1, 2, 1, 1, 1, 1, 1, 1;
0, 0, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1;

Eric Weisstein's World of Mathematics, Dirichlet Divisor Problem

Cf. A000096, A000196, A004526, A005563, A006218, A079643, A094761, A094820, A152948.

(* From Mats Granvik, Feb 21 2016: (Start) *)
nn = 12;
T = Table[
   Sum[Table[
     If[And[If[n*k <= r, If[n >= k, 1, 0], 0] == 1,
       If[(n + 1)*(k + 1) <= r, If[n >= k, 1, 0], 0] == 0], 1, 0], {n,
       1, r}], {k, 1, r}], {r, 1, nn}];
Flatten[T]
A006218a = Table[(n^2 - (2*Sum[Sum[T[[n, k]], {k, 1, kk}], {kk, 1, n}] -
        n)) + 2*n - Round[1 + (1/2)*(-3 + Sqrt[n] + Sqrt[1 + n])], {n,
     1, nn}];
A006218b = -Table[(n^2 - (2*
          Sum[Sum[T[[n, n - k + 1]], {k, 1, kk}], {kk, 1, n}] - n)) -
     2*n + Round[1 + (1/2)*(-3 + Sqrt[n] + Sqrt[1 + n])], {n, 1, nn}];
(A006218b - A006218a);
(* (End) *)
(* From Mats Granvik, May 28 2017: (Start) *)
nn = 12;
T = Table[
   Sum[Table[
     If[And[If[n*k <= r, If[n >= k, 1, 0], 0] == 1,
       If[(n + 1)*(k + 1) <= r, If[n >= k, 1, 0], 0] == 0], 1, 0], {n,
       1, r}], {k, 1, r}], {r, 1, nn}];
Flatten[T]
A006218a = Table[(n^2 - (2*Sum[T[[n, k]]*(n - k + 1), {k, 1, n}] - n)) +
    2*n - Round[1 + (1/2)*(-3 + Sqrt[n] + Sqrt[1 + n])], {n, 1, nn}];
A006218b = Table[-((n^2 - (2*Sum[T[[n, n - k + 1]]*(n - k + 1), {k, 1, n}] -
          n)) - 2*n +
      Round[1 + (1/2)*(-3 + Sqrt[n] + Sqrt[1 + n])]), {n, 1, nn}];
(A006218b - A006218a);
(* (End) *)

See Mathematica program.

cp's OEIS Frontend

A211275 Number of integer pairs (x,y) such that 0 < x <= y <= n and x*y <= floor(n/2).

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A161842 Partial sums of A161841.

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PARI

PARI

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A347517 Partial sums of A347516.

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Mathematica

PARI

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A347527 Partial sums of A347526.

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Mathematica

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PARI

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A212597 Number of ways of writing n in the form i*j+k*m with 0

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Maple

Mathematica

A255315 Lower triangular matrix describing the shape of a half hyperbola in the Dirichlet divisor problem.

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Mathematica

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A212597 Number of ways of writing n in the form ij+km with 0