A010766 -id:A010766 - cp's OEIS frontend

A013942 Triangle of numbers T(n,k) = floor(2n/k), k=1..2n, read by rows.

2, 1, 4, 2, 1, 1, 6, 3, 2, 1, 1, 1, 8, 4, 2, 2, 1, 1, 1, 1, 10, 5, 3, 2, 2, 1, 1, 1, 1, 1, 12, 6, 4, 3, 2, 2, 1, 1, 1, 1, 1, 1, 14, 7, 4, 3, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 16, 8, 5, 4, 3, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 18, 9, 6, 4, 3, 3, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 20, 10, 6, 5, 4, 3

Offset: 1

Clark Kimberling

a(n) is also the leading term in period of continued fraction for n-th nonsquare.

Row A026741(n) contains n and all rows with a smaller row number do not contain n. - Reinhard Zumkeller, Jun 04 2013

			First four rows:
  2 1
  4 2 1 1
  6 3 2 1 1 1
  8 4 2 2 1 1 1 1
  ...

Reinhard Zumkeller, Rows n = 1..100 of table, flattened

Cf. A010766.

Cf. A005843 (row lengths and left edge), A062550 (row sums).

a013942 n k = a013942_tabf !! (n-1) !! (k-1)
a013942_row n = map (div (n * 2)) [1 .. 2 * n]
a013942_tabf = map a013942_row [1 ..]
-- Reinhard Zumkeller, Jun 04 2013

f[n_,h_]:=FractionalPart[(n^2+h)^(1/2)];
g[n_,h_]:=Floor[1/f[n,h]];
TableForm[Table[g[n,h],{n,1,13},{h,1,2n}]]

T(n, k) = 2*n\k;
tabf(nn) = for (n=1, nn, for (k=1, 2*n, print1(T(n,k), ", ")); print()); \\ Michel Marcus, Sep 30 2016

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

T(n,k) = [1/{sqrt(k+n^2)}], k=1,2,...,2n, {}=fractional part, []=floor.

Keyword tabl replaced by tabf and missing a(90)=1 inserted by Reinhard Zumkeller, Jun 04 2013

A123229 Triangle read by rows: T(n, m) = n - (n mod m).

Original entry on oeis.org

1, 2, 2, 3, 2, 3, 4, 4, 3, 4, 5, 4, 3, 4, 5, 6, 6, 6, 4, 5, 6, 7, 6, 6, 4, 5, 6, 7, 8, 8, 6, 8, 5, 6, 7, 8, 9, 8, 9, 8, 5, 6, 7, 8, 9, 10, 10, 9, 8, 10, 6, 7, 8, 9, 10, 11, 10, 9, 8, 10, 6, 7, 8, 9, 10, 11, 12, 12, 12, 12, 10, 12, 7, 8, 9, 10, 11, 12, 13, 12, 12, 12, 10, 12, 7, 8, 9, 10, 11, 12, 13

Offset: 1

Roger L. Bagula and Gary W. Adamson, Oct 06 2006

An equivalent definition: Consider A000012 as a lower-left all-1's triangle, and build the matrix product by multiplication with A127093 from the right. That is, T(n,m) = Sum_{j=m..n} A000012(n,j)*A127093(j,m) = Sum_{j=m..n} A127093(j,m) = m*floor(n/m) = m*A010766(n,m). - Gary W. Adamson, Jan 05 2007

The number of parts k in the triangle is A000203(k) hence the sum of parts k is A064987(k). - Omar E. Pol, Jul 05 2014

			Triangle begins:
{1},
{2, 2},
{3, 2, 3},
{4, 4, 3, 4},
{5, 4, 3, 4, 5},
{6, 6, 6, 4, 5, 6},
{7, 6, 6, 4, 5, 6, 7},
{8, 8, 6, 8, 5, 6, 7, 8},
{9, 8, 9, 8, 5, 6, 7, 8, 9},
...

G. C. Greubel, Rows n=1..100 of triangle, flattened

Cf. A074025, A093960.

Cf. also A024916 (row sums), A127093, A127094, A127096, A127097, A127098, A127099, A038040, A000203, A126988, A127013, A127057.

Flat(List([1..10],n->List([1..n],m->n-(n mod m)))); # Muniru A Asiru, Oct 12 2018

seq(seq(n-modp(n,m),m=1..n),n=1..13); # Muniru A Asiru, Oct 12 2018

a = Table[Table[n - Mod[n, m], {m, 1, n}], {n, 1, 20}]; Flatten[a]

for(n=1,9,for(m=1,n,print1(n-n%m", "))) \\ Charles R Greathouse IV, Nov 07 2011

Edited by N. J. A. Sloane, Jul 05 2014 at the suggestion of Omar E. Pol, who observed that A127095 (Gary W. Adamson, with edits by R. J. Mathar) was the same as this sequence.

A138534 Super least prime signatures; LCM of all signatures with n factors.

Original entry on oeis.org

1, 2, 12, 120, 5040, 110880, 43243200, 1470268800, 1173274502400, 269853135552000, 516498901446528000, 32022931889684736000, 3234636350177055183360000, 265240180714518525035520000, 1163343432613878250805790720000, 6014485546613750556665938022400000

Offset: 0

Alford Arnold, Mar 28 2008

nonn

Also the row product of the following table:
     1
     2
     4   3
     8   3   5
    16   9   5   7
    32   9   5   7   11
    64  27  25   7   11   13
   128  27  25   7   11   13   17
   256  81  25  49   11   13   17   19
   512  81 125  49   11   13   17   19   23
  1024 243 125  49  121   13   17   19   23   29
  ...

			For n = 3 the signatures are {8, 12, 30} so a(3) = 120.

Alois P. Heinz, Table of n, a(n) for n = 0..140
Angelo B. Mingarelli, Abstract factorials, Notes on Number Theory and Discrete Mathematics, Vol. 19, No. 4 (2013), pp. 43-76; arXiv preprint, arXiv:0705.4299 [math.NT], 2007-2012.

Cf. A006218, A010786, A010786, A010766.
Subsequence of A025487.
LCM of terms in rows of A215366.
Cf. A001222, A006218, A346044.

b:= proc(n, i) option remember; `if`(n=0 or i<2, 2^n,
      ilcm(seq(b(n-i*j, i-1)*ithprime(i)^j, j=0..n/i)))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..17);  # Alois P. Heinz, May 15 2015

b[n_, i_] := b[n, i] = If[n == 0 || i < 2, 2^n, LCM @@ Table[b[n - i j, i - 1] Prime[i]^j, {j, 0, n/i}]];
a[n_] := b[n, n];
a /@ Range[0, 17] (* Jean-François Alcover, Nov 02 2020, after Alois P. Heinz *)
a[n_] := Product[Prime[k]^Floor[n/k], {k, 1, n}]; Array[a, 16, 0] (* Amiram Eldar, Jul 02 2021 *)

a(n) = prod(k=1, n, prime(k)^(n\k)); \\ Michel Marcus, Jul 03 2021

From Amiram Eldar, Jul 02 2021: (Start)
a(n) = Product_{k=1..n} prime(k)^floor(n/k).
A001222(a(n)) = A006218(n). (End)
Sum_{n>=0} 1/a(n) = A346044. - Amiram Eldar, Jul 02 2023

More terms from Reikku Kulon, Oct 02 2008

A234575 Triangle T(n, k) read by rows: T(n, k) = floor(n/k) + n mod k.

Original entry on oeis.org

1, 2, 1, 3, 2, 1, 4, 2, 2, 1, 5, 3, 3, 2, 1, 6, 3, 2, 3, 2, 1, 7, 4, 3, 4, 3, 2, 1, 8, 4, 4, 2, 4, 3, 2, 1, 9, 5, 3, 3, 5, 4, 3, 2, 1, 10, 5, 4, 4, 2, 5, 4, 3, 2, 1, 11, 6, 5, 5, 3, 6, 5, 4, 3, 2, 1, 12, 6, 4, 3, 4, 2, 6, 5, 4, 3, 2, 1, 13, 7, 5, 4, 5, 3, 7, 6, 5

Offset: 1

Alex Ratushnyak, Dec 28 2013

			Triangle begins:
   1
   2  1
   3  2  1
   4  2  2  1
   5  3  3  2  1
   6  3  2  3  2  1
   7  4  3  4  3  2  1
   8  4  4  2  4  3  2  1
   9  5  3  3  5  4  3  2  1
  10  5  4  4  2  5  4  3  2  1
  11  6  5  5  3  6  5  4  3  2  1
  12  6  4  3  4  2  6  5  4  3  2  1
  13  7  5  4  5  3  7  6  5  4  3  2  1
  14  7  6  5  6  4  2  7  6  5  4  3  2  1
  15  8  5  6  3  5  3  8  7  6  5  4  3  2  1

Antti Karttunen, Rows n = 1..144 of triangular table, flattened

Cf. A048158, A010766.

a234575 n k = a234575_tabl !! (n-1) !! (k-1)
a234575_row n = a234575_tabl !! (n-1)
a234575_tabl = zipWith (zipWith (+)) a048158_tabl a010766_tabl
-- Reinhard Zumkeller, Apr 29 2015

With[{rows=10},Table[Floor[n/k]+Mod[n,k],{n,rows},{k,n}]] (* Paolo Xausa, Sep 26 2023 *)

for n in range(1, 19):
  for k in range(1, n+1):
    c = n//k + n%k
    print('%2d' % c, end=' ')
  print()

def T(n, k) -> int: return n - (k - 1) * (n // k)
for n in range(1,16): print([T(n, k) for k in range(1,n+1)]) # Peter Luschny, Jun 01 2025

;; MIT/GNU Scheme
(define (A234575bi n k) (+ (floor->exact (/ n k)) (modulo n k)))
(define (A234575 n) (A234575bi (A002024 n) (A002260 n)))
;; Antti Karttunen, Dec 29 2013

T(n, k) = A048158(n, k) + A010766(n, k). - Reinhard Zumkeller, Apr 29 2015
G.f. of the k-th column: x^k*((Sum_{i=0..k-1} x^i) - (k-1)*x^k)/((1 - x)^2*Sum_{i=0..k-1} x^i). - Stefano Spezia, May 08 2024
T(n, k) = n - (k - 1) * floor(n/k). - Peter Luschny, Jun 01 2025

A320224 a(1) = 1; a(n > 1) = Sum_{k = 1..n-1} Sum_{d|k, d < k} a(d).

Original entry on oeis.org

1, 0, 1, 2, 3, 4, 6, 7, 10, 12, 16, 17, 25, 26, 33, 38, 48, 49, 65, 66, 84, 92, 109, 110, 142, 146, 172, 184, 219, 220, 274, 275, 323, 341, 390, 400, 484, 485, 551, 578, 669, 670, 792, 793, 904, 952, 1062, 1063, 1243, 1250, 1408, 1458, 1632, 1633, 1870, 1890

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, A320225.

sol:=[1]; for n in [2..56] do Append(~sol, &+[sol[d]*Floor((n-1)/d-1):d in [1..n-1]]); end for; sol; // Marius A. Burtea, Sep 07 2019

sau[n_]:=If[n==1,1,Sum[sau[d],{k,n-1},{d,Most[Divisors[k]]}]];
Table[sau[n],{n,60}]

seq(n)={my(v=vector(n)); v[1]=1; for(n=2, #v, v[n]=sum(k=1, n-1, v[k]*((n-1)\k - 1))); v} \\ Andrew Howroyd, Sep 07 2019

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

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

A033322 a(n) = floor(2/n).

Original entry on oeis.org

2, 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, 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(2/n): n in [1..100]]; // Wesley Ivan Hurt, Apr 04 2023

Floor[2/Range[80]] (* Wesley Ivan Hurt, Jun 26 2022 *)

A123711 Indices n such that A123709(n) = 8 = number of nonzero terms in row n of triangle A123706.

Original entry on oeis.org

12, 18, 20, 24, 28, 36, 40, 44, 45, 48, 50, 52, 54, 56, 63, 68, 72, 75, 76, 80, 88, 92, 96, 98, 99, 100, 104, 108, 112, 116, 117, 124, 135, 136, 144, 147, 148, 152, 153, 160, 162, 164, 171, 172, 175, 176, 184, 188, 189, 192, 196, 200, 207, 208, 212, 216, 224, 225

Offset: 1

Paul D. Hanna, Oct 09 2006

nonn

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

It appears that this equals A200511, numbers of the form p^k q^m with k,m >= 1, k+m > 2 and p, q prime. - M. F. Hasler, Feb 12 2012

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

Cf. A123706, A123709, A123710, A123712; A010766.

Moebius[i_, j_] := If[Divisible[i, j], MoebiusMu[i/j], 0]; A123709[n_] :=
Length[Select[Table[Moebius[n, j] - Moebius[n, j + 1], {j, 1, n}], # != 0 &]]; Select[Range[500], A123709[#] == 8 &] (* G. C. Greubel, Apr 22 2017 *)

PARI

A123712 Indices n such that 16 = A123709(n) = number of nonzero terms in row n of triangle A123706.

Original entry on oeis.org

60, 84, 90, 120, 126, 132, 140, 150, 156, 168, 180, 198, 204, 220, 228, 234, 240, 252, 260, 264, 270, 276, 280, 294, 300, 306, 308, 312, 315, 336, 340, 342, 348, 350, 360, 364, 372, 378, 380, 396, 408, 414, 440, 444, 450, 456, 460, 468, 476, 480, 490, 492

Offset: 1

Paul D. Hanna, Oct 09 2006

nonn

Triangle A123706 is the matrix inverse of triangle A010766, where A010766(n,k) = [n/k].
a(n) = A178212(n) for n <= 52, possibly more. [Reinhard Zumkeller, May 24 2010]
a(n) = A178212(n) for n <= 2000. - Bill McEachen, Jul 14 2024

G. C. Greubel, Table of n, a(n) for n = 1..1200

Cf. A123706, A123709, A123710, A123711, A010766.

Moebius[i_, j_] := If[Divisible[i, j], MoebiusMu[i/j], 0]; A123709[n_] := Length[Select[Table[Moebius[n, j] - Moebius[n, j + 1], {j, 1, n}], # != 0 &]]; Select[Range[6500], A123709[#] == 16 &] (* G. C. Greubel, Apr 22 2017 *)

is(n)=my(M=matrix(n, n, r, c,r\c)^-1); sum(k=1, n, M[n, k]!=0)==16 \\ Charles R Greathouse IV, Feb 09 2012

A212120 Triangle read by rows T(n,k), n>=1, k>=1, where T(n,k) is the sum of the divisors d of n with min(d, n/d) = k.

Original entry on oeis.org

1, 3, 5, 7, 1, 9, 1, 11, 3, 13, 3, 15, 5, 17, 5, 1, 19, 7, 1, 21, 7, 1, 23, 9, 3, 25, 9, 3, 27, 11, 3, 29, 11, 5, 31, 13, 5, 1, 33, 13, 5, 1, 35, 15, 7, 1, 37, 15, 7, 1, 39, 17, 7, 3, 41, 17, 9, 3, 43, 19, 9, 3, 45, 19, 9, 3, 47, 21, 11, 5, 49, 21, 11, 5, 1

Offset: 1

Omar E. Pol, Jul 02 2012

Column k lists the odd numbers repeated k times starting in row k^2.
1 together with the first differences of the row sums give the divisor function A000005.
T(n,k) is also the total number of divisors of all positive integers <= n on the edges of k-th triangle in the diagram of divisors (see link section). See also A212119.

			Written as an irregular triangle the sequence begins:
1;
3;
5;
7,   1;
9,   1;
11,  3;
13,  3;
15,  5;
17,  5,  1;
19,  7,  1;
21,  7,  1;
23,  9,  3;
25,  9,  3;
27, 11,  3;
29, 11,  5;
31, 13,  5,  1;
33, 13,  5,  1;
35, 15,  7,  1;
37, 15,  7,  1;
39, 17,  7,  3;
41, 17,  9,  3;
43, 19,  9,  3;
45, 19,  9,  3;
47, 21, 11,  5;
49, 21, 11,  5,  1;

Omar E. Pol, Diagram of divisors, figure 1, figure 2.

Row sums give A006218, n >= 1.
Columns (1-5): A005408, A109613, A130823, A129756, A130497.
Cf. A000005, A027750, A010766, A147861, A163100, A212119.

T(n,k) = Sum_{j=1..n} A212119(j,k).

Definition changed by Franklin T. Adams-Watters, Jul 12 2012

A277646 Triangle T(n,k) = floor(n^2/k) for 1 <= k <= n^2, read by rows.

Original entry on oeis.org

1, 4, 2, 1, 1, 9, 4, 3, 2, 1, 1, 1, 1, 1, 16, 8, 5, 4, 3, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 25, 12, 8, 6, 5, 4, 3, 3, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 36, 18, 12, 9, 7, 6, 5, 4, 4, 3, 3, 3, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 49, 24, 16, 12, 9, 8, 7, 6

Offset: 1

Jason Kimberley, Nov 09 2016

			The first five rows of the triangle are:
1;
4, 2, 1, 1;
9, 4, 3, 2, 1, 1, 1, 1, 1;
16, 8, 5, 4, 3, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1;
25, 12, 8, 6, 5, 4, 3, 3, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1;

Jason Kimberley, Table of n, a(n) for n = 1..10416 (the first 31 rows of the triangle)

Cf. Related triangles: A010766, A277647, A277648.
Rows of this triangle (with infinite trailing zeros):
T(1,k) = A000007(k-1),
T(2,k) = A033324(k),
T(3,k) = A033329(k),
T(4,k) = A033336(k),
T(5,k) = A033345(k),
T(6,k) = A033356(k),
T(7,k) = A033369(k),
T(8,k) = A033384(k),
T(9,k) = A033401(k),
T(10,k) = A033420(k),
T(100,k) = A033422(k),
T(10^3,k) = A033426(k),
T(10^4,k) = A033424(k).
Columns of this triangle:
T(n,1) = A000290(n),
T(n,2) = A007590(n),
T(n,3) = A000212(n),
T(n,4) = A002620(n),
T(n,5) = A118015(n),
T(n,6) = A056827(n),
T(n,7) = A056834(n),
T(n,8) = A130519(n+1),
T(n,9) = A056838(n),
T(n,10)= A056865(n),
T(n,12)= A174709(n+2).

A277646:=func;
[A277646(n,k):k in[1..n^2],n in[1..7]];

Table[Floor[n^2/k], {n, 7}, {k, n^2}] // Flatten (* Michael De Vlieger, Nov 24 2016 *)

T(n,k) = A010766(n^2,k).

cp's OEIS Frontend

A013942 Triangle of numbers T(n,k) = floor(2n/k), k=1..2n, read by rows.

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A123229 Triangle read by rows: T(n, m) = n - (n mod m).

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A138534 Super least prime signatures; LCM of all signatures with n factors.

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A234575 Triangle T(n, k) read by rows: T(n, k) = floor(n/k) + n mod k.

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A320224 a(1) = 1; a(n > 1) = Sum_{k = 1..n-1} Sum_{d|k, d < k} a(d).

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A033322 a(n) = floor(2/n).

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A123711 Indices n such that A123709(n) = 8 = number of nonzero terms in row n of triangle A123706.

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