A211159 -id:A211159 - cp's OEIS frontend

A211266 Number of integer pairs (x,y) such that 0

0, 1, 3, 5, 7, 10, 12, 15, 18, 21, 24, 28, 30, 34, 38, 41, 44, 49, 51, 56, 60, 63, 67, 72, 75, 79, 83, 88, 91, 97, 99, 104, 109, 112, 117, 123, 125, 130, 135, 140, 143, 149, 152, 157, 163, 167, 170, 177, 180, 186, 190, 194, 199, 205, 209, 215, 219, 223

Offset: 1

Clark Kimberling, Apr 06 2012

nonn

Guide to related sequences:

A056924 ... 1<=x

A211159 ... 1<=x

A211261 ... 1<=x

A211262 ... 1<=x

A211263 ... 1<=x

A211264 ... 1<=x

A211265 ... 1<=x

A211266 ... 1<=x

A211267 ... 1<=x

A181972 ... 1<=x

A038548 ... 1<=x<=y<=n ... x*y=n

A072670 ... 1<=x<=y<=n ... x*y=n+1

A211270 ... 1<=x<=y<=n ... x*y=2n

A211271 ... 1<=x<=y<=n ... x*y=3n

A211272 ... 1<=x<=y<=n ... x*y=floor(n/2)

A094820 ... 1<=x<=y<=n ... x*y<=n

A091627 ... 1<=x<=y<=n ... x*y<=n+1

A211273 ... 1<=x<=y<=n ... x*y<=2n

A211274 ... 1<=x<=y<=n ... x*y<=3n

A211275 ... 1<=x<=y<=n ... x*y<=floor(n/2)

			a(6) counts these pairs: (1,2), (1,3), (1,4), (1,5), (1,6), (2,3), (2,4), (2,5), (2,6), (3,4).

Cf. A211261, A211264.

a = 1; b = n; z1 = 120;
t[n_] := t[n] = Flatten[Table[x*y, {x, a, b - 1},
{y, x + 1, b}]]
c[n_, k_] := c[n, k] = Count[t[n], k]
Table[c[n, n], {n, 1, z1}]           (* A056924 *)
Table[c[n, n + 1], {n, 1, z1}]       (* A211159 *)
Table[c[n, 2*n], {n, 1, z1}]         (* A211261 *)
Table[c[n, 3*n], {n, 1, z1}]         (* A211262 *)
Table[c[n, Floor[n/2]], {n, 1, z1}]  (* A211263 *)
Print
c1[n_, m_] := c1[n, m] = Sum[c[n, k], {k, a, m}]
Table[c1[n, n], {n, 1, z1}]          (* A211264 *)
Table[c1[n, n + 1], {n, 1, z1}]      (* A211265 *)
Table[c1[n, 2*n], {n, 1, z1}]        (* A211266 *)
Table[c1[n, 3*n], {n, 1, z1}]        (* A211267 *)
Table[c1[n, Floor[n/2]], {n, 1, z1}] (* A181972 *)

A211264 Number of integer pairs (x,y) such that 0 < x < y <= n and x*y <= n.

Original entry on oeis.org

0, 1, 2, 3, 4, 6, 7, 9, 10, 12, 13, 16, 17, 19, 21, 23, 24, 27, 28, 31, 33, 35, 36, 40, 41, 43, 45, 48, 49, 53, 54, 57, 59, 61, 63, 67, 68, 70, 72, 76, 77, 81, 82, 85, 88, 90, 91, 96, 97, 100, 102, 105, 106, 110, 112, 116, 118, 120, 121, 127, 128, 130, 133, 136

Offset: 1

Clark Kimberling, Apr 06 2012

nonn

Partial sums of A056924.

For a guide to related sequences, see A211266.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A211266, A001222, A006218, A000196.

Cf. A066829, A094820, A181972.

[0] cat [&+[(&+[p[2]: p in Factorization(i)] mod 2) *Floor(n div i):i in [2..n] ]:n in [2..65]]; // Marius A. Burtea, Oct 17 2019

with(numtheory): seq(add((bigomega(i) mod 2)*floor(n/i), i=1..n), n=1..60); # Ridouane Oudra, Oct 17 2019
# Alternative:
ListTools:-PartialSums(map(t-> floor(numtheory:-tau(t)/2), [$1..100])); # Robert Israel, Oct 18 2019

a = 1; b = n; z1 = 120;
t[n_] := t[n] = Flatten[Table[x*y, {x, a, b - 1},
{y, x + 1, b}]]
c[n_, k_] := c[n, k] = Count[t[n], k]
Table[c[n, n], {n, 1, z1}]           (* A056924 *)
Table[c[n, n + 1], {n, 1, z1}]       (* A211159 *)
Table[c[n, 2*n], {n, 1, z1}]         (* A211261 *)
Table[c[n, 3*n], {n, 1, z1}]         (* A211262 *)
Table[c[n, Floor[n/2]], {n, 1, z1}]  (* A211263 *)
Print
c1[n_, m_] := c1[n, m] = Sum[c[n, k], {k, a, m}]
Table[c1[n, n], {n, 1, z1}]          (* A211264 *)
Table[c1[n, n + 1], {n, 1, z1}]      (* A211265 *)
Table[c1[n, 2*n], {n, 1, z1}]        (* A211266 *)
Table[c1[n, 3*n], {n, 1, z1}]        (* A211267 *)
Table[c1[n, Floor[n/2]], {n, 1, z1}] (* A181972 *)

from math import isqrt
def A211264(n): return (lambda m: sum(n//k for k in range(1, m+1))-m*(m+1)//2)(isqrt(n)) # Chai Wah Wu, Oct 08 2021

a(n) = (1/2)*Sum_{i=1..n} (1 - A008836(i))*floor(n/i). - Enrique Pérez Herrero, Jul 10 2012 [Corrected by Ridouane Oudra, Oct 17 2019]
From Ridouane Oudra, Oct 17 2019: (Start)
a(n) = Sum_{i=1..n} A066829(i)*floor(n/i)
a(n) = (1/2)*(A006218(n) - A000196(n)). (End)
From Ridouane Oudra, Sep 28 2024: (Start)
a(n) = Sum_{k=1..n} floor((sqrt(k^2 + 4*n) - k)/2) ;
a(n) = A094820(n) - A000196(n) ;
a(n) = A181972(2*n). (End)

A211261 Number of integer pairs (x,y) such that 0

Original entry on oeis.org

0, 0, 1, 1, 1, 2, 1, 1, 2, 2, 1, 3, 1, 2, 3, 2, 1, 3, 1, 3, 3, 2, 1, 4, 2, 2, 3, 3, 1, 5, 1, 2, 3, 2, 3, 5, 1, 2, 3, 4, 1, 5, 1, 3, 5, 2, 1, 5, 2, 3, 3, 3, 1, 5, 3, 4, 3, 2, 1, 7, 1, 2, 5, 3, 3, 5, 1, 3, 3, 5, 1, 6, 1, 2, 5, 3, 3, 5, 1, 5, 4, 2, 1, 7, 3, 2, 3, 4, 1, 8, 3, 3, 3, 2, 3, 6, 1, 3, 5

Offset: 1

Clark Kimberling, Apr 06 2012

nonn

For a guide to related sequences, see A211266.

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A000005, A099777, A211266, A211270.

a = 1; b = n; z1 = 120;
t[n_] := t[n] = Flatten[Table[x*y, {x, a, b - 1},
{y, x + 1, b}]]
c[n_, k_] := c[n, k] = Count[t[n], k]
Table[c[n, n], {n, 1, z1}]           (* A056924 *)
Table[c[n, n + 1], {n, 1, z1}]       (* A211159 *)
Table[c[n, 2*n], {n, 1, z1}]         (* A211261 *)
Table[c[n, 3*n], {n, 1, z1}]         (* A211262 *)
Table[c[n, Floor[n/2]], {n, 1, z1}]  (* A211263 *)
Print
c1[n_, m_] := c1[n, m] = Sum[c[n, k], {k, a, m}]
Table[c1[n, n], {n, 1, z1}]          (* A211264 *)
Table[c1[n, n + 1], {n, 1, z1}]      (* A211265 *)
Table[c1[n, 2*n], {n, 1, z1}]        (* A211266 *)
Table[c1[n, 3*n], {n, 1, z1}]        (* A211267 *)
Table[c1[n, Floor[n/2]], {n, 1, z1}] (* A181972 *)

A211261(n) = sumdiv(2*n,y,(((2*n/y)Antti Karttunen, Sep 30 2018

a(n) = numdiv(n<<1)>>1-1 \\ David A. Corneth, Sep 30 2018

a(n) = floor(A000005(2*n)/2)-1. - Antti Karttunen, Sep 30 2018, after David A. Corneth's PARI-program

A211262 Number of integer pairs (x,y) such that 0

Original entry on oeis.org

0, 0, 0, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 2, 2, 3, 1, 2, 1, 4, 2, 2, 1, 4, 2, 2, 1, 4, 1, 4, 1, 4, 2, 2, 3, 4, 1, 2, 2, 6, 1, 4, 1, 4, 3, 2, 1, 5, 2, 4, 2, 4, 1, 3, 3, 6, 2, 2, 1, 7, 1, 2, 3, 5, 3, 4, 1, 4, 2, 6, 1, 6, 1, 2, 3, 4, 3, 4, 1, 8, 2, 2, 1, 7, 3, 2, 2, 6, 1, 6, 3, 4, 2, 2, 3, 7, 1, 4, 3, 7, 1, 4, 1, 6, 5, 2, 1, 5

Offset: 1

Clark Kimberling, Apr 06 2012

nonn

For a guide to related sequences, see A211266.

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A211266

Cf. also A211271.

a = 1; b = n; z1 = 120;
t[n_] := t[n] = Flatten[Table[x*y, {x, a, b - 1},
 {y, x + 1, b}]]
c[n_, k_] := c[n, k] = Count[t[n], k]
Table[c[n, n], {n, 1, z1}]           (* A056924 *)
Table[c[n, n + 1], {n, 1, z1}]       (* A211159 *)
Table[c[n, 2*n], {n, 1, z1}]         (* A211261 *)
Table[c[n, 3*n], {n, 1, z1}]         (* A211262 *)
Table[c[n, Floor[n/2]], {n, 1, z1}]  (* A211263 *)
Print
c1[n_, m_] := c1[n, m] = Sum[c[n, k], {k, a, m}]
Table[c1[n, n], {n, 1, z1}]          (* A211264 *)
Table[c1[n, n + 1], {n, 1, z1}]      (* A211265 *)
Table[c1[n, 2*n], {n, 1, z1}]        (* A211266 *)
Table[c1[n, 3*n], {n, 1, z1}]        (* A211267 *)
Table[c1[n, Floor[n/2]], {n, 1, z1}] (* A181972 *)

A211262(n) = { my(n3=3*n); sumdiv(n3,d,(d < (n3/d) && (n3/d) <= n)); }; \\ Antti Karttunen, Jan 15 2025

Data section extended up to a(108) by Antti Karttunen, Jan 15 2025

A181972 Number of integer pairs (x,y) such that 0

Original entry on oeis.org

0, 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 6, 6, 7, 7, 9, 9, 10, 10, 12, 12, 13, 13, 16, 16, 17, 17, 19, 19, 21, 21, 23, 23, 24, 24, 27, 27, 28, 28, 31, 31, 33, 33, 35, 35, 36, 36, 40, 40, 41, 41, 43, 43, 45, 45, 48, 48, 49, 49, 53, 53, 54, 54, 57, 57, 59, 59, 61, 61, 63, 63, 67

Offset: 1

Clark Kimberling, Apr 06 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 + 1, b}]]
c[n_, k_] := c[n, k] = Count[t[n], k]
Table[c[n, n], {n, 1, z1}]           (* A056924 *)
Table[c[n, n + 1], {n, 1, z1}]       (* A211159 *)
Table[c[n, 2*n], {n, 1, z1}]         (* A211261 *)
Table[c[n, 3*n], {n, 1, z1}]         (* A211262 *)
Table[c[n, Floor[n/2]], {n, 1, z1}]  (* A211263 *)
Print
c1[n_, m_] := c1[n, m] = Sum[c[n, k], {k, a, m}]
Table[c1[n, n], {n, 1, z1}]          (* A211264 *)
Table[c1[n, n + 1], {n, 1, z1}]      (* A211265 *)
Table[c1[n, 2*n], {n, 1, z1}]        (* A211266 *)
Table[c1[n, 3*n], {n, 1, z1}]        (* A211267 *)
Table[c1[n, Floor[n/2]], {n, 1, z1}] (* A181972 *)

A211263 Number of integer pairs (x,y) such that 0

Original entry on oeis.org

0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 3, 3, 1, 1, 2, 2, 2, 2, 2, 2, 1, 1, 3, 3, 1, 1, 3, 3, 2, 2, 2, 2, 1, 1, 4, 4, 1, 1, 2, 2, 2, 2, 3, 3, 1, 1, 4, 4, 1, 1, 3, 3, 2, 2, 2, 2, 2, 2, 4, 4, 1, 1, 2, 2, 2, 2, 4, 4, 1, 1, 4, 4, 1, 1, 3, 3, 3, 3, 2, 2, 1, 1, 5, 5, 1, 1

Offset: 1

Clark Kimberling, Apr 06 2012

nonn

For a guide to related sequences, see A211266.

			a(12) counts these pairs: (1,6) and (2,3).

Cf. A211266.

a = 1; b = n; z1 = 120;
t[n_] := t[n] = Flatten[Table[x*y, {x, a, b - 1},
 {y, x + 1, b}]]
c[n_, k_] := c[n, k] = Count[t[n], k]
Table[c[n, n], {n, 1, z1}]           (* A056924 *)
Table[c[n, n + 1], {n, 1, z1}]       (* A211159 *)
Table[c[n, 2*n], {n, 1, z1}]         (* A211261 *)
Table[c[n, 3*n], {n, 1, z1}]         (* A211262 *)
Table[c[n, Floor[n/2]], {n, 1, z1}]  (* A211263 *)
Print
c1[n_, m_] := c1[n, m] = Sum[c[n, k], {k, a, m}]
Table[c1[n, n], {n, 1, z1}]          (* A211264 *)
Table[c1[n, n + 1], {n, 1, z1}]      (* A211265 *)
Table[c1[n, 2*n], {n, 1, z1}]        (* A211266 *)
Table[c1[n, 3*n], {n, 1, z1}]        (* A211267 *)
Table[c1[n, Floor[n/2]], {n, 1, z1}] (* A181972 *)

A211265 Number of integer pairs (x,y) such that 0

Original entry on oeis.org

0, 1, 2, 3, 5, 6, 8, 9, 11, 12, 15, 16, 18, 20, 22, 23, 26, 27, 30, 32, 34, 35, 39, 40, 42, 44, 47, 48, 52, 53, 56, 58, 60, 62, 66, 67, 69, 71, 75, 76, 80, 81, 84, 87, 89, 90, 95, 96, 99, 101, 104, 105, 109, 111, 115, 117, 119, 120, 126, 127, 129, 132, 135, 137

Offset: 1

Clark Kimberling, Apr 06 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 + 1, b}]]
c[n_, k_] := c[n, k] = Count[t[n], k]
Table[c[n, n], {n, 1, z1}]           (* A056924 *)
Table[c[n, n + 1], {n, 1, z1}]       (* A211159 *)
Table[c[n, 2*n], {n, 1, z1}]         (* A211261 *)
Table[c[n, 3*n], {n, 1, z1}]         (* A211262 *)
Table[c[n, Floor[n/2]], {n, 1, z1}]  (* A211263 *)
Print
c1[n_, m_] := c1[n, m] = Sum[c[n, k], {k, a, m}]
Table[c1[n, n], {n, 1, z1}]          (* A211264 *)
Table[c1[n, n + 1], {n, 1, z1}]      (* A211265 *)
Table[c1[n, 2*n], {n, 1, z1}]        (* A211266 *)
Table[c1[n, 3*n], {n, 1, z1}]        (* A211267 *)
Table[c1[n, Floor[n/2]], {n, 1, z1}] (* A181972 *)

A211267 Number of integer pairs (x,y) such that 0

Original entry on oeis.org

0, 1, 3, 6, 9, 12, 16, 20, 23, 28, 32, 37, 40, 46, 51, 56, 60, 65, 71, 77, 81, 87, 91, 99, 103, 109, 115, 121, 125, 133, 138, 145, 150, 156, 163, 169, 174, 181, 187, 196, 199, 207, 212, 220, 226, 232, 239, 247, 252, 259, 265, 274, 277, 287, 293, 301, 307

Offset: 1

Clark Kimberling, Apr 06 2012

nonn

For a guide to related sequences, see A211266.

			a(5) counts these pairs: (1,2), (1,3), (1,4), (1,5), (2,3), (2,4), (2,5), (3,4), (3,5).

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A211266.

N:= 100: # for a(1)..a(N)
L:= Vector(N):
for x from 1 to floor(sqrt(N)) do
   for y from x+1 while y<=N and x*y<=3*N do
     n0:= max(y, ceil(x*y/3));
     L[n0]:= L[n0]+1;
od od:
ListTools:-PartialSums(convert(L,list)); # Robert Israel, Oct 18 2019

a = 1; b = n; z1 = 120;
t[n_] := t[n] = Flatten[Table[x*y, {x, a, b - 1},
{y, x + 1, b}]]
c[n_, k_] := c[n, k] = Count[t[n], k]
Table[c[n, n], {n, 1, z1}]           (* A056924 *)
Table[c[n, n + 1], {n, 1, z1}]       (* A211159 *)
Table[c[n, 2*n], {n, 1, z1}]         (* A211261 *)
Table[c[n, 3*n], {n, 1, z1}]         (* A211262 *)
Table[c[n, Floor[n/2]], {n, 1, z1}]  (* A211263 *)
Print
c1[n_, m_] := c1[n, m] = Sum[c[n, k], {k, a, m}]
Table[c1[n, n], {n, 1, z1}]          (* A211264 *)
Table[c1[n, n + 1], {n, 1, z1}]      (* A211265 *)
Table[c1[n, 2*n], {n, 1, z1}]        (* A211266 *)
Table[c1[n, 3*n], {n, 1, z1}]        (* A211267 *)
Table[c1[n, Floor[n/2]], {n, 1, z1}] (* A181972 *)

A347708 Number of distinct possible alternating products of odd-length factorizations of n.

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 1, 3, 1, 1, 2, 2, 1, 2, 1, 3, 1, 1, 1, 3, 1, 1, 1, 3, 1, 2, 1, 2, 2, 1, 1, 4, 1, 2, 1, 2, 1, 2, 1, 3, 1, 1, 1, 5, 1, 1, 2, 3, 1, 2, 1, 2, 1, 2, 1, 5, 1, 1, 2, 2, 1, 2, 1, 4, 2, 1, 1, 5, 1, 1, 1, 3, 1, 3, 1, 2, 1, 1, 1, 5, 1, 2, 2, 3, 1, 2, 1, 3, 2

Offset: 1

Gus Wiseman, Oct 11 2021

nonn

We define the alternating product of a sequence (y_1,...,y_k) to be Product_i y_i^((-1)^(i-1)).
A factorization of n is a weakly increasing sequence of positive integers > 1 with product n.
Note that it is sufficient to look at only length-1 and length-3 factorizations; cf. A347709.

			Representative factorizations for each of the a(180) = 7 alternating products:
  (2*2*3*3*5) -> 5
     (2*2*45) -> 45
     (2*3*30) -> 20
     (2*5*18) -> 36/5
     (2*9*10) -> 20/9
     (3*4*15) -> 45/4
        (180) -> 180

Antti Karttunen, Table of n, a(n) for n = 1..65537

The version for partitions is A028310, reverse A347707.
Positions of 1's appear to be A037143 \ {1}.
The even-length version for n > 1 is A072670, strict A211159.
Counting only integers appears to give A293234, with evens A046951.
This is the odd-length case of A347460, reverse A038548.
The any-length version for partitions is A347461, reverse A347462.
The length-3 case is A347709.
A001055 counts factorizations (strict A045778, ordered A074206).
A056239 adds up prime indices, row sums of A112798.
A276024 counts distinct positive subset-sums of partitions.
A292886 counts knapsack factorizations, by sum A293627.
A301957 counts distinct subset-products of prime indices.
A304792 counts distinct subset-sums of partitions.
A347050 = factorizations w/ an alternating permutation, complement A347706.
A347441 counts odd-length factorizations with integer alternating product.
Cf. A002033, A103919, A108917, A119620, A325770, A339846, A339890, A347437, A347438, A347439, A347440, A347442, A347456.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
altprod[q_]:=Product[q[[i]]^(-1)^(i-1),{i,Length[q]}];
Table[Length[Union[altprod/@Select[facs[n],OddQ[Length[#]]&]]],{n,100}]

altprod(facs) = prod(i=1,#facs,facs[i]^((-1)^(i-1)));
A347708aux(n, m=n, facs=List([])) = if(1==n, if((#facs)%2, altprod(facs), 0), my(newfacs, r, rats=List([])); fordiv(n, d, if((d>1)&&(d<=m), newfacs = List(facs); listput(newfacs,d); r = A347708aux(n/d, d, newfacs); if(r, rats = concat(rats,r)))); (rats));
A347708(n) = if(1==n,0,#Set(A347708aux(n))); \\ Antti Karttunen, Jan 29 2025

Conjecture: For n > 1, a(n) = 1 + A347460(n) - A038548(n) + A072670(n).

Data section extended to a(105) by Antti Karttunen, Jan 29 2025

A200213 Ordered factorizations of n with 2 distinct parts, both > 1.

Original entry on oeis.org

0, 0, 0, 0, 0, 2, 0, 2, 0, 2, 0, 4, 0, 2, 2, 2, 0, 4, 0, 4, 2, 2, 0, 6, 0, 2, 2, 4, 0, 6, 0, 4, 2, 2, 2, 6, 0, 2, 2, 6, 0, 6, 0, 4, 4, 2, 0, 8, 0, 4, 2, 4, 0, 6, 2, 6, 2, 2, 0, 10, 0, 2, 4, 4, 2, 6, 0, 4, 2, 6, 0, 10, 0, 2, 4, 4, 2, 6, 0, 8, 2, 2, 0, 10, 2

Offset: 1

Peter Luschny, Nov 14 2011

			a(24) = 6 = card({{2,12},{3,8},{4,6},{6,4},{8,3},{12,2}}).

Antti Karttunen, Table of n, a(n) for n = 1..10000
Arnold Knopfmacher and Michael Mays, Ordered and Unordered Factorizations of Integers, The Mathematica Journal, Vol 10 (1).

Cf. A000005, A010052, A070824, A161840, A200214, A211159.

a := n -> `if`(n<2, 0, numtheory:-tau(n) - `if`(issqr(n), 3, 2)):
seq(a(n), n = 1..85); # Peter Luschny, Jul 10 2017

OrderedFactorizations[1] = {{}}; OrderedFactorizations[n_?PrimeQ] := {{n}}; OrderedFactorizations[n_] := OrderedFactorizations[n] = Flatten[Function[d, Prepend[#, d] & /@ OrderedFactorizations[n/d]] /@ Rest[Divisors[n]], 1]; a[n_] := With[{of2 = Sort /@ Select[OrderedFactorizations[n], Length[#] == 2 && Length[# // Union] == 2 &] // Union}, Length[Permutations /@ of2 // Flatten[#, 1] &]];  Table[a[n], {n, 1, 85}] (* Jean-François Alcover, Jul 02 2013, copied and adapted from The Mathematica Journal *)

A200213(n) = if(!n,n,sumdiv(n, d, (d<>(n/d))*(d>1)*(dAntti Karttunen, Jul 07 2017

a(n) = if (n==1, 0, numdiv(n) - issquare(n) - 2); \\ Michel Marcus, Jul 07 2017

(define (A200213 n) (if (<= n 1) 0 (- (A000005 n) 2 (A010052 n)))) ;; Antti Karttunen, Jul 07 2017

From Antti Karttunen, Jul 07 & Jul 09 2017: (Start)
a(1) = 0; for n > 1, a(n) = A000005(n) - A010052(n) - 2.
For n >= 2, a(n) = A161840(n) - 2*A010052(n). (End)

Description clarified and term a(0) removed by Antti Karttunen, Jul 09 2017

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A211266 Number of integer pairs (x,y) such that 0

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A211264 Number of integer pairs (x,y) such that 0 < x < y <= n and x*y <= n.

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A211261 Number of integer pairs (x,y) such that 0

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A211262 Number of integer pairs (x,y) such that 0

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A181972 Number of integer pairs (x,y) such that 0

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A211263 Number of integer pairs (x,y) such that 0

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A211265 Number of integer pairs (x,y) such that 0

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A211267 Number of integer pairs (x,y) such that 0

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