A002541 -id:A002541 - cp's OEIS frontend

A348391 Row sums of irregular triangle A348390.

3, 7, 18, 24, 48, 56, 87, 109, 147, 159, 235, 249, 301, 355, 434, 452, 563, 583, 705, 779, 859, 883, 1087, 1143, 1237, 1331, 1499, 1529, 1781, 1813, 2004, 2118, 2240, 2358, 2701, 2739, 2875, 3009, 3339, 3381, 3729, 3773

Offset: 2

Wolfdieter Lang, Nov 07 2021

			a(5) = 2 + 3 + 4 + 5 + 1 + 4 + 1 + 1 + 2 + 1 = (1 + 1 + 1 + 2 + 1) + (2 + 3 + 4 + 5 + 4) = 6 + 18 = 24.

Cf. A002541, A348390.

nterms=50;Table[Total[Flatten[Table[Join[Most[Divisors[k]],Range[2k,n,k]],{k,n}]]],{n,2,nterms+1}] (* Paolo Xausa, Nov 23 2021 *)

a(n) = Sum_{m=1..2*A002541(n)} A348390(n, m), for n >= 2.

a(n) = A153485(n) + A348392(n).

A361660 Irregular triangle read by rows where row n lists the successive numbers moved in the process of forming row n of the triangle A361642.

Original entry on oeis.org

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

Offset: 1

Tamas Sandor Nagy, Mar 19 2023

The first and last numbers in row n>=2 are n and 2, respectively, and they occur just once each in the row.
For row n>=3, and if and only if n-1 is prime, numbers n and 2 are the only numbers which occur just once (since when n-1 is prime it cannot make a rectangle for any other number to move from the initial column to the final row).
A number can move twice in succession, and so occur here twice in succession, when it fills the top right corner cell in a rectangle of width * height = n.
The move is from the initial column to top right corner cell, and therefore the numbers which appear twice in succession are d+1 for each divisor d of n, in the range 1 < d < n.
If n is a prime, then it has no such divisors, or if n is a semiprime n = x*y (including square of a prime) then x+1 and y+1 are the only numbers appearing twice in succession.
The length of row n is A002541(n). This equals to the number of special integer partitions of n there. Where a rectangle is formed of the changing shape, the row length increases more because the movement of a number that completes the rectangle is repeated as it continues to move again.

			The irregular triangle T(n,k) begins:
  n/k |  1  2  3  4  5  6  7  8  9 10 11 12 13 14
  ------------------------------------------------
  1   |  (empty row)
  2   |  2;
  3   |  3, 2;
  4   |  4, 3, 3, 2;
  5   |  5, 4, 3, 4, 2;
  6   |  6, 5, 4, 4, 3, 3, 5, 2;
  7   |  7, 6, 5, 4, 5, 3, 5, 6, 2;
  8   |  8, 7, 6, 5, 5, 4, 6, 3, 3, 4, 7, 2;
  9   |  9, 8, 7, 6, 5, 6, 4, 4, 7, 3, 7, 6, 8, 2;
.
Movements of the six-number-high column. 1 never moves. 4 and 3 move twice each in immediate succession as 6 is a composite and a semiprime:
.
  6
  5   5
  4   4     4
  3   3     3     3 4   3
  2   2     2 5   2 5   2 5     2 5 3   2 5       2
  1   1 6   1 6   1 6   1 6 4   1 6 4   1 6 4 3   1 6 4 3 5   1 6 4 3 5 2
.
The parallel is shown for row length and the special integer partition in A002541:
For n = 4, its row consists of 4, 3, 3 and 2, that is four elements.
The special partition of n = 4 is (4), (2 2), (3 1), and (2 1 1), that is also four partitions. The relation is demonstrated by the illustration below. Square blocks represent the four numbers. As they move, the changing shape assumes a number of identical or reflected formations. The number of possible grouping of the blocks within them is exactly the same as the number of the moves that the blocks undergo:
.  _ _
  |   |__________ 1st move
  |   |     _ _  |
  |   |    |   |_|____________ 2nd move ____________________________ 4th move
  |   |    |   | |       _ _ _v_      _|_                           |
  |   |    |   | |      |   |   |____|___|_____ 3rd move            |
  |   |    |   |_v_     |   |   |    |   |_ _ _v_      _ _ _ _ _ _ _v_
  |   |    |   |   |    |   |   |    |   |   |   |    |               |
  |_ _|    |_ _|_ _|    |_ _|_ _|    |_ _|_ _|_ _|    |_ _ _ _ _ _ _ _|
    4        3   1        2   2        2   1   1              4
    ^                                                         ^
    |____________________ Identical partition ________________|

Cf. A361642, A002541 (row length).

A004199 Table of [ x/y ], where (x,y) = (1,1),(1,2),(2,1),(1,3),(2,2),(3,1),...

Original entry on oeis.org

1, 0, 2, 0, 1, 3, 0, 0, 1, 4, 0, 0, 1, 2, 5, 0, 0, 0, 1, 2, 6, 0, 0, 0, 1, 1, 3, 7, 0, 0, 0, 0, 1, 2, 3, 8, 0, 0, 0, 0, 1, 1, 2, 4, 9, 0, 0, 0, 0, 0, 1, 1, 2, 4, 10, 0, 0, 0, 0, 0, 1, 1, 2, 3, 5, 11, 0, 0, 0, 0, 0, 0, 1, 1, 2, 3, 5, 12, 0, 0, 0, 0, 0, 0, 1, 1, 1, 2, 3, 6, 13, 0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 2, 4, 6, 14

Offset: 1

David W. Wilson

Entry in row n and column k is also the number of multiples of k less than or equal to n, n,k >= 1. - L. Edson Jeffery, Aug 31 2014

			Array begins:
  1, 0, 0, 0, 0, 0, 0, 0, ...
  2, 1, 0, 0, 0, 0, 0, 0, ...
  3, 1, 1, 0, 0, 0, 0, 0, ...
  4, 2, 1, 1, 0, 0, 0, 0, ...
  5, 2, 1, 1, 1, 0, 0, 0, ...
  ...

Cf. A002541 (antidiagonal sums).

Cf. A010766 (same sequence as triangle, omitting the zeros), A010783.

(* Array version: *)
Grid[Table[Floor[n/k], {n, 14}, {k, 14}]] (* L. Edson Jeffery, Aug 31 2014 *)
(* Array antidiagonals flattened: *)
Flatten[Table[Floor[(n - k + 1)/k], {n, 14}, {k, n}]] (* L. Edson Jeffery, Aug 31 2014 *)

Sum_{k=1..n} a(n-k+1,k) = A002541(n+1).

A144489 Partial sums of A087624.

Original entry on oeis.org

0, 0, 0, 1, 1, 3, 3, 4, 5, 7, 7, 9, 9, 11, 13, 14, 14, 16, 16, 18, 20, 22, 22, 24, 25, 27, 28, 30, 30, 33, 33, 34, 36, 38, 40, 42, 42, 44, 46, 48, 48, 51, 51, 53, 55, 57, 57, 59, 60, 62, 64, 66, 66, 68, 70, 72, 74, 76, 76, 79, 79, 81, 83, 84, 86, 89, 89, 91, 93, 96, 96, 98, 98, 100, 102

Offset: 1

N. J. A. Sloane, Dec 11 2008

nonn

Number of pairs (p,q) with 1 < p < q <= n, p|q and p prime. - Wesley Ivan Hurt, Oct 30 2022

Cf. A002541, A087624.

Accumulate[Array[If[PrimeQ[#],0,PrimeNu[#]]&,110]] (* Harvey P. Dale, Jul 19 2019 *)

a(n) = sum(k=1, n, if (!isprime(k), omega(k))); \\ Michel Marcus, Nov 06 2022

A248517 Number of odd divisors > 1 in the numbers 1 through n, counted with multiplicity.

Original entry on oeis.org

0, 0, 0, 1, 1, 2, 3, 4, 4, 6, 7, 8, 9, 10, 11, 14, 14, 15, 17, 18, 19, 22, 23, 24, 25, 27, 28, 31, 32, 33, 36, 37, 37, 40, 41, 44, 46, 47, 48, 51, 52, 53, 56, 57, 58, 63, 64, 65, 66, 68, 70, 73, 74, 75, 78, 81, 82, 85, 86, 87, 90, 91, 92, 97, 97, 100, 103, 104, 105, 108, 111

Offset: 0

R. J. Mathar, Jun 18 2015

Number of partitions of n into 3 parts such that the smallest part divides the "middle" part. - Wesley Ivan Hurt, Oct 21 2021

Charles R Greathouse IV, Table of n, a(n) for n = 0..10000

Cf. A002541, A006218, A069283.

A248517 := proc(n)
    add(A069283(j),j=1..n) ;
end proc:

Table[Sum[Floor[Floor[i/2]/(n - i)], {i, n - 1}], {n, 0, 100}] (* Wesley Ivan Hurt, Jan 30 2016 *)
Join[{0},Accumulate[Table[Count[Divisors[n],?OddQ]-1,{n,80}]]] (* _Harvey P. Dale, Jan 06 2019 *)
Join[{0}, Accumulate[Table[DivisorSigma[0, n/2^IntegerExponent[n, 2]] - 1, {n, 1, 100}]]] (* Amiram Eldar, Jul 10 2022 *)

a(n)=my(n2=n\2); sum(k=1, sqrtint(n), n\k)*2-sqrtint(n)^2-sum(k=1, sqrtint(n2), n2\k)*2+sqrtint(n2)^2-n \\ Charles R Greathouse IV, Jun 18 2015

from math import isqrt
def A248517(n): return ((t:=isqrt(m:=n>>1))+(s:=isqrt(n)))*(t-s)+(sum(n//k for k in range(1,s+1))-sum(m//k for k in range(1,t+1))<<1)-n # Chai Wah Wu, Oct 23 2023

a(n) = Sum_{j=1..n} A069283(j).
a(n) = A060831(n) - n.
a(n) = A006218(n) - A006218(floor(n/2)) - n. - Charles R Greathouse IV, Jun 18 2015
a(n) = Sum_{i=1..n-1} floor(floor(i/2)/(n-i)). - Wesley Ivan Hurt, Jan 30 2016

A330476 a(n) = Sum_{m=2..n} floor(n/m)^2.

Original entry on oeis.org

0, 1, 2, 6, 7, 16, 17, 28, 34, 47, 48, 75, 76, 93, 108, 134, 135, 172, 173, 212, 231, 256, 257, 322, 332, 361, 384, 435, 436, 513, 514, 571, 598, 635, 658, 760, 761, 802, 833, 926, 927, 1028, 1029, 1104, 1165, 1214, 1215, 1358, 1372, 1453, 1492, 1579, 1580, 1705, 1736, 1857, 1900, 1961, 1962

Offset: 1

Robert Israel, Dec 15 2019

nonn

			a(4) = floor(4/2)^2 + floor(4/3)^2 + floor(4/4)^2 = 6.

Robert Israel, Table of n, a(n) for n = 1..5000

Cf. A002541, A224526.

[0] cat [&+[Floor(n/m)^2:m in [2..n]]:n in [2..60]]; // Marius A. Burtea, Dec 15 2019

f:=  proc(n) local m; add(floor(n/m)^2,m=2..n) end proc:
map(f, [$1..100]);

Table[Sum[IntegerPart[(n/m)]^2,{m,2,n}],{n,1,100}] (* Metin Sariyar, Dec 15 2019 *)

A333885 Number of triples (i,j,k) with 1 <= i < j < k <= n such that i divides j divides k.

Original entry on oeis.org

0, 0, 0, 1, 1, 3, 3, 6, 7, 9, 9, 16, 16, 18, 20, 26, 26, 33, 33, 40, 42, 44, 44, 59, 60, 62, 65, 72, 72, 84, 84, 94, 96, 98, 100, 119, 119, 121, 123, 138, 138, 150, 150, 157, 164, 166, 166, 192, 193, 200, 202, 209, 209, 224, 226, 241, 243, 245, 245, 276, 276

Offset: 1

Derek Lim, Apr 08 2020

nonn

			The a(4) = 1 triple is (1,2,4).
The a(8) = 6 triples are (1,2,4), (1,2,6), (1,2,8), (1,3,6), (1,4,8), (2,4,8).

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

Cf. A000005, A002541, A320224.

with(numtheory):
a:= proc(n) option remember; `if`(n=0, 0, a(n-1)+
       add(tau(d)-1, d=divisors(n) minus {n}))
    end:
seq(a(n), n=1..80);  # Alois P. Heinz, Apr 09 2020

a[n_] := a[n] = If[n == 0, 0, a[n - 1] + Sum[DivisorSigma[0, d] - 1, {d, Most @ Divisors[n]}]];
Array[a, 80] (* Jean-François Alcover, Nov 27 2020, after Alois P. Heinz *)

an = len([(i,j,k) for i in range(1,n+1) for j in range(i+1,n+1) for k in range(j+1,n+1) if j%i==0 and k%j==0])

a(n) = Sum_{m=1..n} Sum_{d|m, dAlois P. Heinz, Apr 09 2020

A339804 a(n) = Sum_{k=1..floor(n/2)} k * (n-k) * floor((n-k)/k).

Original entry on oeis.org

0, 1, 4, 13, 22, 50, 68, 116, 162, 236, 278, 437, 498, 634, 794, 1018, 1118, 1450, 1574, 1975, 2276, 2598, 2774, 3519, 3834, 4273, 4746, 5490, 5772, 6887, 7214, 8163, 8856, 9586, 10330, 12072, 12540, 13443, 14382, 16244, 16806, 18861, 19480, 21192, 22954, 24267

Offset: 1

Wesley Ivan Hurt, Dec 17 2020

Total volume of all rectangular prisms with dimensions (x, y, z) where x and y are positive integers such that x + y = n, x <= y, and z = floor(y/x). - Wesley Ivan Hurt, Dec 20 2020

Cf. A002541, A153485, A279847, A339370.

Table[Sum[k (n - k)*Floor[(n - k)/k], {k, Floor[n/2]}], {n, 50}]

a(n) = sum(k=1, n\2, k*(n-k)*((n-k)\k)); \\ Michel Marcus, Dec 19 2020

from math import isqrt
def A339804(n): return (n*(1-n**2)+((s:=isqrt(n))**4<<1)+s**3*(3*(1-n))+s**2*(1-3*n) + sum((q:=n//k)*(-6*k**2+n*(3*((k<<1)+q+1))-q*((q<<1)+3)-1) for k in range(1,s+1)))//6 # Chai Wah Wu, Oct 27 2023

a(n) ~ n^3*(Pi^2-2-4*zeta(3))/12. - Rok Cestnik, Dec 19 2020

a(n) = n*A153485(n) - A279847(n). - Vaclav Kotesovec, Dec 21 2020

A361819 Irregular triangle read by rows where T(n,k) is the distance which number A361660(n,k) moves in the process described in A361642.

Original entry on oeis.org

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

Offset: 1

Tamas Sandor Nagy, Mar 25 2023

Number A361660(n,k) moves to the right and then down and T(n,k) counts the steps in both.
All moves are T(n,k) >= 2 steps since a number moves at least one step right and one step down.
Row n has sum A002378(n-1) which is the total steps to move a column down to a row irrespective of the order of movement.
Each row is a palindrome (the same when reversed), since the moves in A361642 are exactly the reverse moves to send its row back to the starting column.

			Irregular triangle T(n,k) begins:
  n/k     |   1    2    3    4    5    6    7    8    9
  ------------------------------------------------------
  1       |   (empty row)
  2       |   2;
  3       |   3,   3;
  4       |   4,   2,   2,   4;
  5       |   5,   3,   4,   3,   5;
  6       |   6,   4,   2,   3,   3,   2,   4,   6;
  7       |   7,   5,   3,   5,   2,   5,   3,   5,   7;
 ...

Cf. A361642, A361660, A002541 (row lengths), A002378 (row sums).

function a = A361819( max_row )
    k = 1;
    for r = 2:max_row
        h = zeros(1,r); h(1) = r;
        while max(h) > 1
           j =  find(h == max(h), 1, 'last' );
           m =  find(h < max(h)-1, 1, 'first' );
           a(k) = (m-j) + (h(j)-h(m)) - 1;
           h(j) = h(j) - 1; h(m) = h(m) + 1;
           k = k+1;
        end
    end
end % Thomas Scheuerle, Mar 27 2023

A113240 Expansion of (1/(1-x))*sum(k>=2,x^k/(1-2x^k)).

Original entry on oeis.org

1, 2, 5, 6, 13, 14, 25, 30, 49, 50, 97, 98, 165, 186, 325, 326, 621, 622, 1161, 1230, 2257, 2258, 4481, 4498, 8597, 8858, 17125, 17126, 34077, 34078, 66985, 68014, 133553, 133634, 267057, 267058, 529205, 533306, 1058261, 1058262, 2115133

Offset: 0

Paul Barry, Oct 19 2005

Cf. A002541.

a(n)=sum{k=0..floor(n/2), floor((n-k+1)/(k+1))*2^k}.

cp's OEIS Frontend

A348391 Row sums of irregular triangle A348390.

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A361660 Irregular triangle read by rows where row n lists the successive numbers moved in the process of forming row n of the triangle A361642.

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A004199 Table of [ x/y ], where (x,y) = (1,1),(1,2),(2,1),(1,3),(2,2),(3,1),...

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A144489 Partial sums of A087624.

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A248517 Number of odd divisors > 1 in the numbers 1 through n, counted with multiplicity.

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A330476 a(n) = Sum_{m=2..n} floor(n/m)^2.

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A333885 Number of triples (i,j,k) with 1 <= i < j < k <= n such that i divides j divides k.

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A339804 a(n) = Sum_{k=1..floor(n/2)} k * (n-k) * floor((n-k)/k).

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