A010766 -id:A010766 - cp's OEIS frontend

A320225 a(1) = 1; a(n > 1) = Sum_{k = 1..n} Sum_{d|k, d < k} a(d).

1, 1, 2, 4, 5, 9, 10, 16, 19, 26, 27, 44, 45, 57, 65, 87, 88, 120, 121, 158, 171, 200, 201, 278, 284, 331, 353, 426, 427, 536, 537, 646, 676, 766, 782, 982, 983, 1106, 1154, 1365, 1366, 1617, 1618, 1851, 1943, 2146, 2147, 2589, 2600, 2917, 3008, 3390, 3391

Offset: 1

Gus Wiseman, Oct 07 2018

nonn

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

Cf. A002541, A003238, A010766, A014668, A126656, A167865, A316782, A317712, A320222, A320224.

sau[n_]:=If[n==1,1,Sum[sau[d],{k,n},{d,Most[Divisors[k]]}]];
Table[sau[n],{n,30}]

from functools import lru_cache
@lru_cache(maxsize=None)
def A320225(n): return 1 if n == 1 else sum(A320225(d)*(n//d-1) for d in range(1,n)) # Chai Wah Wu, Jun 08 2022

a(1) = 1; a(n > 1) = Sum_{d = 1..n-1} a(d) * floor(n/d-1).

G.f. A(x) satisfies A(x) = x + (1/(1 - x)) * Sum_{k>=2} A(x^k). - Ilya Gutkovskiy, Sep 06 2019

A123707 a(n) = Sum_{k=1..n} A123706(n,k)*2^(k-1).

Original entry on oeis.org

1, 0, 1, 3, 7, 14, 31, 60, 126, 248, 511, 1005, 2047, 4064, 8183, 16320, 32767, 65394, 131071, 261885, 524255, 1048064, 2097151, 4193220, 8388600, 16775168, 33554304, 67104765, 134217727, 268427002, 536870911, 1073725440, 2147483135

Offset: 1

Paul D. Hanna, Oct 09 2006

nonn

Triangle A123706 is the matrix inverse of triangle A010766(n,k) = [n/k].

G. C. Greubel, Table of n, a(n) for n = 1..1000

Cf. A123706, A123708, A123709.

t[n_, k_] := If[Divisible[n, k], MoebiusMu[n/k], 0] - If[Divisible[n, k + 1], MoebiusMu[n/(k + 1)], 0]; Table[Sum[t[n, k]*2^(k - 1), {k, 1, n}], {n, 1, 50}] (* G. C. Greubel, Oct 26 2017 *)

{a(n)=sum(k=1,n,(matrix(n,n,r,c,if(r>=c,floor(r/c)))^-1)[n,k]*2^(k-1))}

G.f.: Sum_{k>=1} mu(k) * x^k * (1 - x^k) / (1 - 2*x^k). - Ilya Gutkovskiy, Feb 06 2020

a(n) ~ 2^(n-2). - Vaclav Kotesovec, May 03 2025

A242114 Triangle read by rows: T(n,k) = number of pairs (x,y) in {1..n}X{1..n} with gcd(x,y) = k.

Original entry on oeis.org

1, 3, 1, 7, 1, 1, 11, 3, 1, 1, 19, 3, 1, 1, 1, 23, 7, 3, 1, 1, 1, 35, 7, 3, 1, 1, 1, 1, 43, 11, 3, 3, 1, 1, 1, 1, 55, 11, 7, 3, 1, 1, 1, 1, 1, 63, 19, 7, 3, 3, 1, 1, 1, 1, 1, 83, 19, 7, 3, 3, 1, 1, 1, 1, 1, 1, 91, 23, 11, 7, 3, 3, 1, 1, 1, 1, 1, 1, 115, 23

Offset: 1

Reinhard Zumkeller, May 04 2014

T(n,1) = A018805(n);
sum(T(n,k): k = 1..n) = A000290(n);
sum(T(n,k): k = 2..n) = A100613(n);
T(floor(n/k),1) = A018805(n).

			T(4,1) = #{(1,1), (1,2), (1,3), (1,4), (2,1), (2,3), (3,1), (3,2), (3,4), (4,1), (4,3)} = 11;
T(4,2) = #{(2,2), (2,4), (4,2)} = 3;
T(4,3) = #{(3,3)} = 1;
T(4,4) = #{(4,4)} = 1.
The triangle begins:                                            row sums
.   1:    1                                                            1
.   2:    3   1                                                        4
.   3:    7   1   1                                                    9
.   4:   11   3   1   1                                               16
.   5:   19   3   1   1  1                                            25
.   6:   23   7   3   1  1  1                                         36
.   7:   35   7   3   1  1  1  1                                      49
.   8:   43  11   3   3  1  1  1  1                                   64
.   9:   55  11   7   3  1  1  1  1  1                                81
.  10:   63  19   7   3  3  1  1  1  1  1                            100
.  11:   83  19   7   3  3  1  1  1  1  1  1                         121
.  12:   91  23  11   7  3  3  1  1  1  1  1  1                      144
.  13:  115  23  11   7  3  3  1  1  1  1  1  1  1                   169
.  14:  127  35  11   7  3  3  3  1  1  1  1  1  1  1                196
.  15:  143  35  19   7  7  3  3  1  1  1  1  1  1  1  1             225
.  16:  159  43  19  11  7  3  3  3  1  1  1  1  1  1  1  1          256
.  17:  191  43  19  11  7  3  3  3  1  1  1  1  1  1  1  1  1       289
.  18:  203  55  23  11  7  7  3  3  3  1  1  1  1  1  1  1  1  1    324 .

Reinhard Zumkeller, Rows n = 1..125 of table, flattened

a242114 n k = a242114_tabl !! (n-1) !! (k-1)
a242114_row n = a242114_tabl !! (n-1)
a242114_tabl = map (map a018805) a010766_tabl

T[n_, k_] := 2 Total[EulerPhi[Range[Quotient[n, k]]]] - 1;
Table[T[n, k], {n, 1, 18}, {k, 1, n}] // Flatten (* Jean-François Alcover, Sep 20 2021 *)

T(n,k) = A018805(A010766(n,k));

A350103 Triangle read by rows. Number of self-measuring subsets of the initial segment of the natural numbers strictly below n and cardinality k. Number of subsets S of [n] with S = distset(S) and |S| = k.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 1, 1, 4, 2, 1, 1, 1, 1, 5, 2, 1, 1, 1, 1, 1, 6, 3, 2, 1, 1, 1, 1, 1, 7, 3, 2, 1, 1, 1, 1, 1, 1, 8, 4, 2, 2, 1, 1, 1, 1, 1, 1, 9, 4, 3, 2, 1, 1, 1, 1, 1, 1, 1, 10, 5, 3, 2, 2, 1, 1, 1, 1, 1, 1, 1, 11, 5, 3, 2, 2, 1, 1, 1, 1, 1, 1

Offset: 0

Peter Luschny, Dec 14 2021

We use the notation [n] = {0, 1, ..., n-1}. If S is a subset of [n] then we define the distset of S (set of distances of S) as {|x - y|: x, y in S}. We call a subset S of the natural numbers self-measuring if and only if S = distset(S).

			Triangle starts:
[ 0] [1]
[ 1] [1, 1]
[ 2] [1, 1,  1]
[ 3] [1, 1,  2, 1]
[ 4] [1, 1,  3, 1, 1]
[ 5] [1, 1,  4, 2, 1, 1]
[ 6] [1, 1,  5, 2, 1, 1, 1]
[ 7] [1, 1,  6, 3, 2, 1, 1, 1]
[ 8] [1, 1,  7, 3, 2, 1, 1, 1, 1]
[ 9] [1, 1,  8, 4, 2, 2, 1, 1, 1, 1]
[10] [1, 1,  9, 4, 3, 2, 1, 1, 1, 1, 1]
[11] [1, 1, 10, 5, 3, 2, 2, 1, 1, 1, 1, 1]
[12] [1, 1, 11, 5, 3, 2, 2, 1, 1, 1, 1, 1, 1]
.
The first  column is 1,1,...  because {} = distset({}) and |{}| = 0.
The second column is 1,1,... because {0} = distset({0}) and |{0}| = 1.
The third  column is n-1  because {0, j} = distset({0, j}) and |{0, j}| = 2 for j = 1..n - 1.
The main diagonal is 1,1,... because [n] = distset([n]) and |[n]| = n (these are the complete rulers A103295).

Winston de Greef, Table of n, a(n) of the first 150 rows, flattened (n = 0..11324)
Peter Luschny, Illustrating self-measuring subsets of {0, 1, 2, 3}.

Cf. A350102 (row sums), A349976, A103295, A003988, A010766, A123706.

T := (n, k) -> ifelse(k < 2, 1, floor((n - 1) / (k - 1))):
seq(print(seq(T(n, k), k = 0..n)), n = 0..12);

distSet[s_] := Union[Map[Abs[Differences[#][[1]]] &, Union[Sort /@ Tuples[s, 2]]]]; T[n_, k_] := Count[Subsets[Range[0, n - 1]], ?((ds = distSet[#]) == # && Length[ds] == k &)]; Table[T[n, k], {n, 0, 12}, {k, 0, n}] // Flatten (* _Amiram Eldar, Dec 16 2021 *)

T(n, k) = if(k<=1, 1, (n - 1) \ (k - 1)) \\ Winston de Greef, Jan 31 2024

# generating and counting (slow)
def isSelfMeasuring(R):
    S, L = Set([]), len(R)
    R = Set([r - 1 for r in R])
    for i in range(L):
        for j in (0..i):
            S = S.union(Set([abs(R[i] - R[i - j])]))
    return R == S
def A349976row(n):
    counter = [0] * (n + 1)
    for S in Subsets(n):
        if isSelfMeasuring(S): counter[len(S)] += 1
    return counter
for n in range(10): print(A349976row(n))

T(n, k) = floor((n - 1) / (k - 1)) for k >= 2.

T(n, k) = 1 if k = 0 or k = 1 or n >= k >= floor((n + 1)/2).

A010783 Triangle of numbers floor(n/(n-k)).

Original entry on oeis.org

1, 1, 2, 1, 1, 3, 1, 1, 2, 4, 1, 1, 1, 2, 5, 1, 1, 1, 2, 3, 6, 1, 1, 1, 1, 2, 3, 7, 1, 1, 1, 1, 2, 2, 4, 8, 1, 1, 1, 1, 1, 2, 3, 4, 9, 1, 1, 1, 1, 1, 2, 2, 3, 5, 10, 1, 1, 1, 1, 1, 1, 2, 2, 3, 5, 11, 1, 1, 1, 1, 1, 1, 2, 2, 3, 4, 6, 12

Offset: 1

N. J. A. Sloane

Alois P. Heinz, Table of n, a(n) for n = 1..10011

Cf. A010766 (mirrored).

a010783 n k = (n + 1 - k) `div` k
a010783_row n = a010783_tabl !! (n-1)
a010783_tabl = map reverse a010766_tabl
-- Reinhard Zumkeller, Apr 29 2015

T:= (n,k)-> floor(n/(n-k)):
seq (seq (T(n, k), k=0..n-1), n=1..15); # Alois P. Heinz, Jul 15 2011

A033330 a(n) = floor(10/n).

Original entry on oeis.org

10, 5, 3, 2, 2, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 1

Jeff Burch

Index entries for linear recurrences with constant coefficients, signature (1).

Cf. A010766.

[Floor(10/n): n in [1..100]]; // Wesley Ivan Hurt, Apr 04 2023

Floor[10/Range[100]] (* Harvey P. Dale, Jun 12 2019 *)

A075993 Triangle read by rows: T(n,m) is the number of integers k such that floor(n/k) = m, n >= 1, k = 1..n.

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Sep 28 2002

The sum of numbers in row n is n.

Number of terms > 0 per row: Sum_{k=1..n} A057427(T(n,k)) = A055086(n). - Reinhard Zumkeller, Apr 06 2006

			T(5, 1) = 3 counts k such that floor(5/k) = 1, namely k = 5, 4, 3.
First 10 rows:
  1
  1 1
  2 0 1
  2 1 0 1
  3 1 0 0 1
  3 1 1 0 0 1
  4 1 1 0 0 0 1
  4 2 0 1 0 0 0 1
  5 1 1 1 0 0 0 0 1
  5 2 1 0 1 0 0 0 0 1

Michael De Vlieger, Table of n, a(n) for n = 1..11325 (rows 1 <= n <= 150, flattened)

Columns 1, 2, 3 are essentially A004526, A008615, A008679.

Cf. A010766.

Table[Floor[n/m] - Floor[n/(m + 1)], {n, 14}, {m, n}] // Flatten (* Michael De Vlieger, Jan 14 2022 *)

T(n, m) = floor(n/m) - floor(n/(m+1)).

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

Original entry on oeis.org

1, 4, 8, 12, 17, 21, 27, 31, 37, 42, 48, 52, 60, 64, 70, 76, 83, 87, 95, 99, 107, 113, 119, 123, 133, 138, 144, 150, 158, 162, 172, 176, 184, 190, 196, 202, 213, 217, 223, 229, 239, 243, 253, 257, 265, 273, 279, 283, 295, 300, 308, 314, 322, 326, 336, 342, 352

Offset: 0

Ralf Stephan, May 06 2005

Number of ordered triples [k,l,m] with n = k+l*m and k, l, m all in the range [0..n].
From R. J. Mathar, Jun 30 2013: (Start)
A010766 is the following array A read by antidiagonals:
  1,  1,  1,  1,  1,  1, ...
  2,  1,  1,  1,  1,  1, ...
  3,  2,  1,  1,  1,  1, ...
  4,  2,  2,  1,  1,  1, ...
  5,  3,  2,  2,  1,  1, ...
  6,  3,  2,  2,  2,  1, ...
and apparently a(n) is the hook sum Sum_{k=0..n} A(n,k) + Sum_{r=0..n-1} A(r,n). (End)

			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.
a(3)=12: 3+0*0, 3+0*m (6), 2+1*1, 1+2*1 (2), 0+3*1 (2).

R. J. Mathar, Table of n, a(n) for n = 0..1000

Cf. A006218, A106634, A106846, A106847.

Cf. A005408, A006218.

A106633 := proc(n)
    local a, k, l, m ;
    a := 0 ;
    for k from 0 to n do
        for l from 0 to n do
            if l = 0 then
                if k = n then
                    a := a+n+1 ;
                end if;
            else
                m := (n-k)/l ;
                if type(m,'integer') then
                    a := a+1 ;
                end if;
            end if;
        end do:
    end do:
    a ;
end proc: # R. J. Mathar, Oct 17 2012

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];
Table[A106633[n], {n, 0, 56}] (* Jean-François Alcover, Jun 10 2023, after R. J. Mathar *)

list(n)={
    my(v=vector(n),t);
    for(i=2,n,for(j=1,min(n\i,i-1),v[i*j]+=2));
    for(i=1,sqrtint(n),v[i^2]++);
    concat(1,vector(n,k,2*k+1+t+=v[k]))
}; \\ Charles R Greathouse IV, Oct 17 2012

From Ridouane Oudra, Apr 22 2024: (Start)
a(n) = 2*n + 1 + Sum_{k=1..n} floor(n/k);
a(n) = 2*n + 1 + Sum_{k=1..n} tau(k);
a(n) = A005408(n) + A006218(n). (End)

Definition clarified by N. J. A. Sloane, Jul 07 2012

A115725 Number of partitions with maximum rectangle <= n.

Original entry on oeis.org

1, 2, 5, 10, 26, 42, 118, 171, 389, 692, 1442, 1854, 5534, 6895, 11910, 21116, 44278, 52568, 118734, 138670, 300326, 492507, 728514, 829244, 2167430, 2987124, 4167602, 6092588, 11308432, 12554900, 29925267, 33023589, 57950313, 81424281, 106214784, 148101088

Offset: 0

Franklin T. Adams-Watters, Mar 11 2006

nonn

A partition has maximum rectangle <= n iff it is a subpartition of row n of A010766.

			The 10 partitions with maximum rectangle <= 3: 0: []; 1: [1]; 2: [2], [1^2], [2,1]; 3: [3], [1^3], [3,1], [2,1^2], [3,1^2].

Alois P. Heinz, Table of n, a(n) for n = 0..100
Eric Weisstein's World of Mathematics, Ferrers Diagram.

Cf. A115728, A115729, A115724, A010766.

a(n) = subpart([A115728 (or A115729), [] is row n of A010766.

a(n) = Sum_{k>=0} A182114(k,n). - Alois P. Heinz, Nov 02 2012

A123708 a(n) = sum of unsigned elements in row n of triangle A123706.

Original entry on oeis.org

1, 3, 3, 4, 3, 5, 3, 4, 4, 7, 3, 8, 3, 7, 7, 4, 3, 8, 3, 8, 7, 7, 3, 8, 4, 7, 4, 8, 3, 13, 3, 4, 7, 7, 7, 8, 3, 7, 7, 8, 3, 13, 3, 8, 8, 7, 3, 8, 4, 8, 7, 8, 3, 8, 7, 8, 7, 7, 3, 16, 3, 7, 8, 4, 7, 13, 3, 8, 7, 15, 3, 8, 3, 7, 8, 8, 7, 13, 3, 8, 4, 7, 3, 16, 7, 7, 7, 8, 3, 16, 7, 8, 7, 7, 7, 8, 3, 8, 8, 8, 3, 13

Offset: 1

Paul D. Hanna, Oct 09 2006

nonn

Triangle A123706 is the matrix inverse of triangle A010766, where A010766(n,k) = [n/k].

G. C. Greubel, Table of n, a(n) for n = 1..1000

Cf. A123706, A123707, A123709; A010766.

t[n_, k_] := If[Divisible[n, k], MoebiusMu[n/k], 0] - If[Divisible[n, k + 1], MoebiusMu[n/(k + 1)], 0]; Table[Sum[Abs[t[n, k]], {k, 1, n}], {n, 1, 50}] (* G. C. Greubel, Oct 26 2017 *)

{a(n)=local(M=matrix(n,n,r,c,if(r>=c,floor(r/c)))^-1);sum(k=1,n,abs(M[n,k]))}

cp's OEIS Frontend

A320225 a(1) = 1; a(n > 1) = Sum_{k = 1..n} Sum_{d|k, d < k} a(d).

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A123707 a(n) = Sum_{k=1..n} A123706(n,k)*2^(k-1).

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A242114 Triangle read by rows: T(n,k) = number of pairs (x,y) in {1..n}X{1..n} with gcd(x,y) = k.

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A350103 Triangle read by rows. Number of self-measuring subsets of the initial segment of the natural numbers strictly below n and cardinality k. Number of subsets S of [n] with S = distset(S) and |S| = k.

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A010783 Triangle of numbers floor(n/(n-k)).

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A033330 a(n) = floor(10/n).

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A075993 Triangle read by rows: T(n,m) is the number of integers k such that floor(n/k) = m, n >= 1, k = 1..n.

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

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