A002541 -id:A002541 - cp's OEIS frontend

A330926 a(n) = Sum_{k=1..n} (ceiling(n/k) mod 2).

1, 1, 2, 1, 3, 2, 3, 2, 5, 3, 4, 3, 6, 5, 6, 3, 7, 6, 7, 6, 9, 6, 7, 6, 11, 9, 10, 7, 10, 9, 10, 9, 14, 11, 12, 9, 13, 12, 13, 10, 15, 14, 15, 14, 17, 12, 13, 12, 19, 17, 18, 15, 18, 17, 18, 15, 20, 17, 18, 17, 22, 21, 22, 17, 23, 20, 21, 20, 23, 20, 21, 20, 27, 26, 27, 22, 25, 22, 23, 22

Offset: 1

Ilya Gutkovskiy, May 25 2020

nonn

a(n) = number of terms among {ceiling(n/k)}, 1 <= k <= n, that are odd.

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

Cf. A002541, A006218, A048272, A059851, A075997, A120885, A320226, A325939, A332682, A333194.

b:= n-> add((-1)^d, d=numtheory[divisors](n)):
a:= proc(n) option remember; `if`(n>0, 1+b(n-1)+a(n-1), 0) end:
seq(a(n), n=1..80);  # Alois P. Heinz, May 25 2020

Table[Sum[Mod[Ceiling[n/k], 2], {k, 1, n}], {n, 1, 80}]
Table[n - Sum[DivisorSum[k, (-1)^(# + 1) &], {k, 1, n - 1}], {n, 1, 80}]
nmax = 80; CoefficientList[Series[x/(1 - x) (1 + Sum[x^(2 k)/(1 + x^k), {k, 1, nmax}]), {x, 0, nmax}], x] // Rest

a(n) = sum(k=1, n, ceil(n/k) % 2); \\ Michel Marcus, May 25 2020

from math import isqrt
def A330926(n): return n+(s:=isqrt(n-1))**2-((t:=isqrt(m:=n-1>>1))**2<<1)-(sum((n-1)//k for k in range(1,s+1))-(sum(m//k for k in range(1,t+1))<<1)<<1) # Chai Wah Wu, Oct 23 2023

G.f.: (x/(1 - x)) * (1 + Sum_{k>=1} x^(2*k) / (1 + x^k)).
a(n) = n - Sum_{k=1..n-1} A048272(k).
a(n) = A075997(n-1) + 1.

A348389 Irregular triangle read by rows: row n gives for n >= 2 a concatenation of the finite sequences of the multiples of k, larger than k and not exceeding n, for k = 1, 2, ..., floor(n/2).

Original entry on oeis.org

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

Offset: 2

Wolfdieter Lang, Oct 31 2021

The length of row n is A002541(n).
The sum of row n is A348392(n).
The lengths of the sublists for these multiples of k in row n are given by T(n, k) = A348388(n, k), for n >= 2 and k = 1, 2, ..., floor(n/2).

			The irregular triangle a(n, m) begins: (the k-sublists are separated by a vertical bar)
n\m   1 2 3 4 5 6 7 8  9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 ...
-------------------------------------------------------------------------
2:    2
3:    2 3
4:    2 3 4|4
5:    2 3 4 5|4
6:    2 3 4 5 6|4 6|6
7:    2 3 4 5 6 7|4 6| 6
8:    2 3 4 5 6 7 8|4  6  8| 6| 8
9:    2 3 4 5 6 7 8 9| 4  6  8| 6  9| 8
10:   2 3 4 5 6 7 8 9 10| 4  6  8 10| 6  9| 8|10
11:   2 3 4 5 6 7 8 9 10 11| 4  6  8 10| 6  9| 8|10
12:   2 3 4 5 6 7 8 9 10 11 12| 4  6  8 10 12| 6  9 12| 8 12|10|12
13:   2 3 4 5 6 7 8 9 10 11 12 13| 4  6  8 10 12| 6  9 12| 8 12|10|12
...

Cf. A002541, A348388, A348392.

nrows=10;Table[Flatten[Table[Range[2k,n,k],{k,Floor[n/2]}]],{n,2,nrows+1}] (* Paolo Xausa, Nov 23 2021 *)

The entries a(n, m) of row n, for n > = 2 and m = 1, 2, ..., A002541(n), are given by the concatenation of the sequences k*(2, 3, ..., t(n,k)) for k = 1, 2, ..., floor(n/2), with t(n, k) = floor((n-k)/k) + 1.

A348390 Irregular triangle read by rows: for n >= 2 the row members a(n, m) give the proper divisors of k, followed by the multiples of k larger than k and not exceeding n, for k = 1, 2, ..., n.

Original entry on oeis.org

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

Offset: 2

Wolfdieter Lang, Nov 07 2021

The length of row n is 2*A002541(n), for n >= 2.
The sum of row n is A348391(n). The sum of the proper divisors of row n is A153485(n). The sum of the multiples in row n is A348392(n). Hence, A348391(n) =  A153485(n) + A348392(n).
For k = 1 the proper divisor set is empty, and for k > floor(n/2) the set of multiples is empty.

			The irregular triangle a(n, m), m = 1, 2, ..., 2*A002541(n) begins:
(members for k = 1, 2, ..., n are separated by a vertical bar, and the proper divisors and multiples are separated by a comma)
n\m 1 2 3 4 5 6 7 8  9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 ...
-----------------------------------------------------------------------------------
2:  2|1
3:  2 3|1|1
4:  2 3 4|1,4|1|1 2
5:  2 3 4 5|1,4|1|1  2| 1
6:  2 3 4 5 6|1,4 6| 1, 6| 1  2| 1| 1 2 3
7:  2 3 4 5 6 7|1,4  6| 1, 6| 1  2| 1| 1  2  3| 1
8:  3 4 5 6 7 8|1,4  6  8| 1 ,6| 1  2 ,8| 1| 1  2  3| 1| 1  2  4
9:  2 3 4 5 6 7 8 9| 1, 4  6  8| 1, 6  9| 1  2, 8| 1| 1  2  3| 1| 1  2  4| 1  3
...
n = 10: 2 3 4 5 6 7 8 9 10 | 1, 4 6 8 10 | 1, 6 9 | 1 2, 8 | 1, 10 | 1 2 3 | 1 | 1 2 4 | 1 3 | 1 2 5
-----------------------------------------------------------------------------------
n = 4:  d(4, 1) = {}, m(4, 1) = {2, 3, 4}; d(4, 2) = {1}, m(4, 2) = {4}; d(4, 3) = {1}, m(4, 3) = {}; d(4, 4) = {1, 2}, m(4, 4) = {}, This explains row n = 4.

Cf. A002541, A027751, A153485, A348389, A348391, A348392.

nrows=10;Table[Flatten[Table[Join[Most[Divisors[k]],Range[2k,n,k]],{k,n}]],{n,2,nrows+1}] (* Paolo Xausa, Nov 23 2021 *)

For n >= 2 row n gives the sequence of the sequence d(n, k) of proper divisors of k (A027751(k)) followed by the sequences m(n, k) of the multiples of k, larger than k and not exceeding n (A348389), for k = 1, 2, 3, ..., n.

A348392 Row sums of the irregular triangle A348389.

Original entry on oeis.org

2, 5, 13, 18, 36, 43, 67, 85, 115, 126, 186, 199, 241, 286, 350, 367, 457, 476, 576, 639, 705, 728, 896, 946, 1024, 1105, 1245, 1274, 1484, 1515, 1675, 1774, 1876, 1981, 2269, 2306, 2420, 2537, 2817

Offset: 2

Wolfdieter Lang, Oct 31 2021

			n = 4: Compare the row of an array with all multiples of k <= n, for k = 1,2, ..., n with the row of A348389:
All multiples of k <= 4 for k = 1..4:  [1 2 3 4|2 4|3|4] with row sum A143127(4) = 23.
A348389 row 4: [2 3 4|4] with 1, 2, 3 and 4 missing: row sum is 23 - 4*5/2  = 13. Hence a(4) = A143127(4) - A000217(4).
Also: a(4) =  A348391(4) - A153485(4) = 18 - 5 = 13.

Cf. A000217, A002541, A143127, A348389.

a(n) = Sum_{m=1.. A002541(n)} A348389(n, m), for n >= 2.
a(n) = A143127(n) - A000217(n).
a(n) = A348391(n) - A153485(n).

A366967 a(n) = Sum_{k=2..n} binomial(k,2) * floor(n/k).

Original entry on oeis.org

0, 1, 4, 11, 21, 40, 61, 96, 135, 191, 246, 337, 415, 528, 646, 801, 937, 1145, 1316, 1568, 1802, 2089, 2342, 2737, 3047, 3451, 3841, 4338, 4744, 5358, 5823, 6474, 7060, 7758, 8384, 9294, 9960, 10835, 11657, 12717, 13537, 14739, 15642, 16881, 18025, 19314, 20395

Offset: 1

Seiichi Manyama, Oct 30 2023

nonn

Partial sums of A069153.
Cf. A002541, A236632.
Cf. A024916, A064602, A366971.

a(n) = sum(k=2, n, binomial(k, 2)*(n\k));

from math import isqrt
def A366967(n): return ((s:=isqrt(n))**2*(1-s**2)+sum((q:=n//k)*(3*k*(k-1)+q**2-1) for k in range(1,s+1)))//6 # Chai Wah Wu, Oct 30 2023

G.f.: 1/(1-x) * Sum_{k>=1} x^(2*k)/(1-x^k)^3 = 1/(1-x) * Sum_{k>=2} binomial(k,2) * x^k/(1-x^k).

a(n) = (A064602(n)-A024916(n))/2. - Chai Wah Wu, Oct 30 2023

A224525 Number of idempotent 3 X 3 0..n matrices of rank 1.

Original entry on oeis.org

27, 69, 123, 195, 273, 375, 477, 603, 735, 885, 1035, 1221, 1395, 1593, 1803, 2031, 2253, 2511, 2757, 3039, 3321, 3615, 3909, 4251, 4575, 4917, 5271, 5649, 6015, 6429, 6819, 7245, 7671, 8109, 8559, 9051, 9513, 9999, 10497, 11031, 11541, 12099, 12633

Offset: 1

R. H. Hardin, Apr 09 2013

nonn

Row 3 of A224524.

			Some solutions for n=3:
  0 1 0   1 3 2   0 1 0   1 3 1   0 0 0   0 0 0   1 0 0
  0 1 0   0 0 0   0 1 0   0 0 0   0 1 2   3 0 3   1 0 0
  0 3 0   0 0 0   0 2 0   0 0 0   0 0 0   1 0 1   2 0 0

Robert Israel, Table of n, a(n) for n = 1..10000 (first 80 terms from R. H. Hardin)

Cf. A002541, A224524.

f:= k -> 6*k^2 + 18*k + 3 + 6 * add(floor(k/m),m=2..k):
map(f, [$1..50]); # Robert Israel, Dec 15 2019

a(n) = 6*n^2 + 18*n + 3 + 6 * A002541(n). - Robert Israel, Dec 15 2019

A224526 Number of idempotent 4 X 4 0..n matrices of rank 1.

Original entry on oeis.org

108, 404, 892, 1716, 2732, 4324, 6060, 8516, 11308, 14820, 18572, 23668, 28716, 34916, 41836, 49860, 58076, 68164, 78252, 90356, 102988, 116868, 131276, 148564, 165660, 184532, 204604, 226788, 249116, 274900, 300252, 328628, 357868, 389028, 421580, 457924, 493500

Offset: 1

R. H. Hardin, Apr 09 2013

nonn

Row 4 of A224524

			Some solutions for n=3:
  0 0 0 0     0 0 0 0     1 0 0 0     0 0 0 0     0 0 0 0
  0 0 0 0     2 0 0 1     0 0 0 0     1 1 0 0     0 0 0 0
  2 0 1 0     2 0 0 1     0 0 0 0     3 3 0 0     0 0 0 0
  0 0 0 0     2 0 0 1     0 0 0 0     2 2 0 0     1 0 1 1

Robert Israel, Table of n, a(n) for n = 1..4000

Cf. A002541, A024917, A224524.

F4 := k -> 8*k^3 + 36*k^2 + 24*add(m*floor(k/m), m = 2 .. k) + 12*add(floor(k/m), m = 2 .. k) + 12*add(floor(k/m)^2, m = 2 .. k) + 60*k + 4:
map(F4, [$1..100]); # Robert Israel, Dec 15 2019

Table[8*n^3+36*n^2+60*n+4+24*Sum[k*Floor[n/k],{k, 2, n}]+12*Sum[Floor[(n-k)/k],{k, n-1}]+12*Sum[Floor[(n/k)]^2,{k,2,n}],{n,1,100}] (* Metin Sariyar, Dec 15 2019 *)

a(n) = 8*n^3 + 36*n^2 + 60*n + 4 + 24*A024917(n) + 12*A002541(n) + 12*Sum_{m=2..n} floor(n/m)^2. - Robert Israel, Dec 15 2019

More terms from Metin Sariyar, Dec 15 2019

A338991 a(n) = Sum_{k=1..floor(n/2)} (n-2k) floor((n-k)/k).

Original entry on oeis.org

0, 0, 2, 6, 13, 24, 37, 56, 78, 106, 132, 178, 212, 258, 312, 376, 425, 508, 565, 662, 749, 836, 909, 1058, 1156, 1264, 1384, 1536, 1636, 1836, 1946, 2126, 2282, 2434, 2606, 2880, 3019, 3194, 3385, 3676, 3833, 4138, 4305, 4572, 4863, 5086, 5271, 5692, 5924, 6240

Offset: 1

Wesley Ivan Hurt, Dec 21 2020

Total area of all rectangles with dimensions (y-x) X (z) where x and y are integers such that x + y = n, 0 < x <= y, and z = floor(y/x).

Cf. A001620, A002541, A006218, A024916, A153485, A279847, A339370.

Table[Sum[(n - 2 k)*Floor[(n - k)/k], {k, Floor[n/2]}], {n, 60}]

from math import isqrt
def A338991(n): return ((s:=isqrt(n))+1)*(n*(1-s)+s**2)-sum((q:=n//k)*((k-n<<1)+q+1) for k in range(1,s+1)) # Chai Wah Wu, Oct 23 2023

From Vaclav Kotesovec, Jun 24 2021: (Start)
a(n) = n + n*A006218(n) - 2*A024916(n).
a(n) ~ (log(n) + 2*gamma - Pi^2/6 - 1)*n^2, where gamma is the Euler-Mascheroni constant A001620. (End)

A339217 a(n) = Sum_{k=1..n} floor((2*n-k)/k).

Original entry on oeis.org

0, 1, 4, 8, 12, 17, 23, 27, 34, 40, 46, 52, 60, 65, 73, 81, 87, 93, 104, 108, 118, 126, 132, 140, 150, 157, 165, 173, 183, 189, 201, 205, 216, 226, 232, 242, 254, 258, 268, 278, 288, 295, 307, 313, 323, 335, 343, 349, 363, 369, 382, 390, 398, 408, 420, 428, 440, 448, 456, 464, 482

Offset: 0

Wesley Ivan Hurt, Dec 22 2020

Cf. A002541, A153485, A279847, A338991, A339370.

Table[Sum[Floor[(2 n - i)/i], {i, n}], {n, 0, 60}]

a(n) = sum(k=1, n, (2*n-k)\k); \\ Michel Marcus, Dec 22 2020

From Vaclav Kotesovec, Dec 23 2020: (Start)
For n>0, a(n) = 2*A006218(n) + A075989(n) - n.
a(n) ~ 2*n * (log(2*n) + 2*gamma - 2), where gamma is the Euler-Mascheroni constant A001620. (End)

A348388 Irregular triangle read by rows: T(n, k) = floor((n-k)/k), for k = 1, 2, ..., floor(n/2) and n >= 2.

Original entry on oeis.org

1, 2, 3, 1, 4, 1, 5, 2, 1, 6, 2, 1, 7, 3, 1, 1, 8, 3, 2, 1, 9, 4, 2, 1, 1, 10, 4, 2, 1, 1, 11, 5, 3, 2, 1, 1, 12, 5, 3, 2, 1, 1, 13, 6, 3, 2, 1, 1, 1, 14, 6, 4, 2, 2, 1, 1, 15, 7, 4, 3, 2, 1, 1, 1, 16, 7, 4, 3, 2, 1, 1, 1, 17, 8, 5, 3, 2, 2, 1, 1, 1, 18, 8, 5, 3, 2, 2, 1, 1, 1, 19, 9, 5, 4, 3, 2, 1, 1, 1, 1

Offset: 2

Wolfdieter Lang, Oct 31 2021

This irregular triangle T(n, k) gives the number of multiples of number k, larger than k and not exceeding n, for k = 1, 2, ..., floor(n/2), for n >= 2. See A348389 for the array of these multiples.
The length of row n is floor(n/2) = A004526(n), for n >= 2.
The row sums give A002541(n). See the formula given there by Wesley Ivan Hurt, May 08 2016.
The columns give the k-fold repeated positive integers k, for k >= 1.

			The irregular triangle T(n, k) begins:
n\k   1 2 3 4 5 6 7 8 9 10 ...
------------------------------
2:    1
3:    2
4:    3 1
5:    4 1
6:    5 2 1
7:    6 2 1
8:    7 3 1 1
9:    8 3 2 1
10:   9 4 2 1 1
11:  10 4 2 1 1
12:  11 5 3 2 1 1
13:  12 5 3 2 1 1
14:  13 6 3 2 1 1 1
15:  14 6 4 2 2 1 1
16:  15 7 4 3 2 1 1 1
17:  16 7 4 3 2 1 1 1
18:  17 8 5 3 2 2 1 1 1
19:  18 8 5 3 2 2 1 1 1
20:  19 9 5 4 3 2 1 1 1  1
...

Karl-Heinz Hofmann, Table of n, a(n) for n = 2..10101

Columns k (with varying offsets): A000027, A004526, A008620, A008621, A002266, A097992, ...

Cf. A002541, A004526, A010766, A348389.

T[n_, k_] := Floor[(n - k)/k]; Table[T[n, k], {n, 2, 20}, {k, 1, Floor[n/2]}] // Flatten (* Amiram Eldar, Nov 02 2021 *)

def A348388row(n): return [(n - k) // k for k in range(1, 1 + n // 2)]
for n in range(2, 21): print(A348388row(n))  # Peter Luschny, Nov 05 2021

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

G.f. of column k: G(k, x) = x^(2*k)/((1 - x)*(1 - x^k)).

cp's OEIS Frontend

A330926 a(n) = Sum_{k=1..n} (ceiling(n/k) mod 2).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Python

Formula

A348389 Irregular triangle read by rows: row n gives for n >= 2 a concatenation of the finite sequences of the multiples of k, larger than k and not exceeding n, for k = 1, 2, ..., floor(n/2).

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula

A348390 Irregular triangle read by rows: for n >= 2 the row members a(n, m) give the proper divisors of k, followed by the multiples of k larger than k and not exceeding n, for k = 1, 2, ..., n.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula

A348392 Row sums of the irregular triangle A348389.

Views

Author

Keywords

Examples

Crossrefs

Formula

A366967 a(n) = Sum_{k=2..n} binomial(k,2) * floor(n/k).

Views

Author

Keywords

Crossrefs

Programs

PARI

Python

Formula

A224525 Number of idempotent 3 X 3 0..n matrices of rank 1.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Formula

A224526 Number of idempotent 4 X 4 0..n matrices of rank 1.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

Extensions

A338991 a(n) = Sum_{k=1..floor(n/2)} (n-2*k) * floor((n-k)/k).

Views

Author

Keywords

Comments

Crossrefs

A338991 a(n) = Sum_{k=1..floor(n/2)} (n-2k) floor((n-k)/k).