A004736 -id:A004736 - cp's OEIS frontend

A056534 Mapping from the ordering by product (A027750, A056538) to the ordering by sum (A002260, A004736) of ordered pairs (a,b), a>=1, b>=1.

1, 2, 3, 4, 6, 7, 5, 10, 11, 15, 16, 8, 9, 21, 22, 28, 29, 12, 14, 36, 37, 13, 45, 46, 17, 20, 55, 56, 66, 67, 23, 18, 19, 27, 78, 79, 91, 92, 30, 35, 105, 106, 24, 26, 120, 121, 38, 25, 44, 136, 137, 153, 154, 47, 31, 34, 54, 171, 172, 190, 191, 57, 32, 33, 65, 210, 211, 39

Offset: 1

Antti Karttunen, Jun 20 2000

nonn

			The "ordering by sum": (1,1),(1,2),(2,1),(1,3),(2,2),(3,1),(1,4),(2,3),(3,2),(4,1),...
The "ordering by product": (1,1),(1,2),(2,1),(1,3),(3,1),(1,4),(2,2),(4,1),(1,5),(5,1),...

Inverse: A056535.

ordered_pair_perm := proc(upto_n) local a,i,j; a := []; for i from 1 to upto_n do for j in sort(divisors(i)) do a := [op(a),binomial(((i/j) + j - 1),2)+j]; od; od; RETURN(a); end;

max = 21; A056534 = {}; For[i = 1, i <= max, i++, Do[ AppendTo[ A056534, Binomial[i/j + j - 1, 2] + j], {j, Divisors[i]}]]; A056534 (* Jean-François Alcover, Oct 05 2012, after Maple *)

A158823 Triangle read by rows: matrix product A004736 * A158821.

Original entry on oeis.org

1, 3, 1, 6, 2, 2, 10, 3, 4, 3, 15, 4, 6, 6, 4, 21, 5, 8, 9, 8, 5, 28, 6, 10, 12, 12, 10, 6, 36, 7, 12, 15, 16, 15, 12, 7, 45, 8, 14, 18, 20, 20, 18, 14, 8, 55, 9, 16, 21, 24, 25, 24, 21, 16, 9, 66, 10, 18, 24, 28, 30, 30, 28, 24, 18, 10, 78, 11, 20, 27, 32, 35, 36, 35, 32, 27, 20, 11

Offset: 1

Gary W. Adamson & Roger L. Bagula, Mar 28 2009

			First few rows of the triangle =
   1;
   3,  1;
   6,  2,  2;
  10,  3,  4,  3;
  15,  4,  6,  6,  4;
  21,  5,  8,  9,  8,  5;
  28,  6, 10, 12, 12, 10,  6;
  36,  7, 12, 15, 16, 15, 12,  7;
  45,  8, 14, 18, 20, 20, 18, 14,  8;
  55,  9, 16, 21, 24, 25, 24, 21, 16,  9;
  66, 10, 18, 24, 28, 30, 30, 28, 24, 18, 10;
  78, 11, 20, 27, 32, 35, 36, 35, 32, 27, 20, 11;
  91, 12, 22, 30, 36, 40, 42, 42, 40, 36, 30, 22, 12;

G. C. Greubel, Rows n = 1..50 of the triangle, flattened

Cf. A000292 (row sums), A003991, A004736, A158821.

[k eq 1 select Binomial(n+1, 2) else (n-k+1)*(k-1): k in [1..n], n in [1..15]]; // G. C. Greubel, Apr 01 2021

A158823 := proc(n,m) add( A004736(n,k)*A158821(k-1,m-1),k=1..n) ; end: seq(seq(A158823(n,m),m=1..n),n=1..8) ; # R. J. Mathar, Oct 22 2009

Table[If[k==1, Binomial[n+1, 2], (n-k+1)*(k-1)], {n,15}, {k,n}]//Flatten (* G. C. Greubel, Apr 01 2021 *)

flatten([[binomial(n+1, 2) if k==1 else (n-k+1)*(k-1) for k in (1..n)] for n in (1..15)]) # G. C. Greubel, Apr 01 2021

Sum_{k=1..n} T(n, k) = A000292(n).
T(n, k) = Sum_{j=k..n} A004736(n, j)*A158821(j-1, k-1).
From R. J. Mathar, Mar 03 2011: (Start)
T(n, k) = (n-k+1)*(k-1), k>1.
T(n, 1) = A000217(n). (End)

Corrected A-number in a formula - R. J. Mathar, Oct 30 2009

A195107 Fractalization of the fractal sequence A004736. Interspersion fractally induced by A004736.

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Sep 09 2011

nonn

See A194959 for a discussion of fractalization and the interspersion fractally induced by a sequence. The sequence A004736 is the fractal sequence obtained by concatenating the segments 1; 2,1; 3,2,1; 4,3,2,1;...

Cf. A194959, A004736, A195108, A195109.

j[n_] := Table[n + 1 - k, {k, 1, n}]; t[1] = j[1];
t[n_] := Join[t[n - 1], j[n]]   (* A004736 *)
t[10]
p[n_] := t[20][[n]]
g[1] = {1}; g[n_] := Insert[g[n - 1], n, p[n]]
f[1] = g[1]; f[n_] := Join[f[n - 1], g[n]]
f[20] (* A195107 *)
row[n_] := Position[f[30], n];
u = TableForm[Table[row[n], {n, 1, 5}]]
v[n_, k_] := Part[row[n], k];
w = Flatten[Table[v[k, n - k + 1], {n, 1, 13},
{k, 1, n}]] (* A195108 *)
q[n_] := Position[w, n]; Flatten[Table[q[n],
{n, 1, 80}]] (* A195109 *)

A093919 Consider the triangle in A004736, the k-th term of the n-th row is the LCM of the preceding k terms.

Original entry on oeis.org

1, 2, 2, 3, 6, 6, 4, 12, 12, 12, 5, 20, 60, 60, 60, 6, 30, 60, 60, 60, 60, 7, 42, 210, 420, 420, 420, 420, 8, 56, 168, 840, 840, 840, 840, 840, 9, 72, 504, 504, 2520, 2520, 2520, 2520, 2520, 10, 90, 360, 2520, 2520, 2520, 2520, 2520, 2520, 2520, 11, 110, 990, 3960

Offset: 1

Amarnath Murthy, Apr 25 2004

			T(5,3) = 60 because the fifth row of A004736 has the terms {5 4 3 2 1 }, the first three terms being 5, 4 & 3 which have an LCM of 60.
Triangle begins:
1
2 2
3 6 6
4 12 12 12
5 20 60 60 60
6 30 60 60 60 60
7 42 210 420 420 420 420
8 56 168 840 840 840 840 840

Enrique Pérez Herrero, Rows n=1..100 of triangle, flattened.

Cf. A004736, A093920, A179661.

T[n_, k_] := LCM @@ Drop[Reverse[Range[n]], (k - n)]; Flatten[ Table[ T[n, k], {n, 11}, {k, n}]] (* Robert G. Wilson v, Apr 27 2004 *)

Edited by Robert G. Wilson v, Apr 27 2004

A173395 a(n) = (A002260(n) + 1) * (A004736(n) + 1).

Original entry on oeis.org

4, 6, 6, 8, 9, 8, 10, 12, 12, 10, 12, 15, 16, 15, 12, 14, 18, 20, 20, 18, 14, 16, 21, 24, 25, 24, 21, 16, 18, 24, 28, 30, 30, 28, 24, 18, 20, 27, 32, 35, 36, 35, 32, 27, 20, 22, 30, 36, 40, 42, 42, 40, 36, 30, 22, 24, 33, 40, 45, 48, 49, 48, 45, 40, 33, 24, 26

Offset: 1

Fabio Civolani (civox(AT)tiscali.it), Feb 17 2010

Every number of this sequence is composite, and every composite number appears in this sequence.

Viewed as a square array this sequence is the multiplication table with headers starting at 2: A002260 and A004736 being indexing functions for square arrays, a(n)=T(i,j) with i=A002260(n) and j=A004736(n), T(i,j)=(i+1)(j+1). - Luc Rousseau, Oct 15 2017

			4;
6,6;
8,9,8;
10,12,12,10;
12,15,16,15,12;
From _Luc Rousseau_, Oct 15 2017: (Start)
Viewed as a square array,
   4  6  8 10 12 ...
   6  9 12 15 18 ...
   8 12 16 20 24 ...
  10 15 20 25 30 ...
  12 18 24 30 36 ...
  ...
= the multiplication table with headers starting at 2.
(End)

Michael De Vlieger, Table of n, a(n) for n = 1..11325 (rows 1 <= n <= 150).
Michael Somos, Sequences used for indexing triangular or square arrays.

Cf. A002260, A004736.

Map[Times @@ # & /@ Transpose@{#, Reverse@ #} &, Array[Range, 12] + 1] // Flatten (* Michael De Vlieger, Oct 16 2017 *)

a(n) = ((2*n + round(sqrt(2*n)) - round(sqrt(2*n))^2)/2 + 1)*((2 - 2*n + round(sqrt(2*n)) + round(sqrt(2*n))^2)/2 + 1) \\ Michel Marcus, Jun 19 2013

a(n)=my(s=round(sqrt(n*=2)));(n-s-s^2-4)*(n+s-s^2+2)/4 \\ Charles R Greathouse IV, Jun 19 2013

a(n) = ((2 n + round(sqrt(2n)) - round(sqrt(2n))^2)/2 + 1)((2 - 2n + round(sqrt(2n)) + round(sqrt(2n))^2)/2 + 1).

A195108 Interspersion fractally induced by A004736.

Original entry on oeis.org

1, 2, 3, 5, 6, 4, 8, 10, 7, 9, 13, 15, 11, 14, 12, 19, 21, 17, 20, 18, 16, 26, 28, 23, 27, 24, 22, 25, 34, 36, 30, 35, 32, 29, 33, 31, 43, 45, 39, 44, 41, 37, 42, 40, 38, 53, 55, 49, 54, 51, 47, 52, 50, 48, 46, 64, 66, 59, 65, 62, 57, 63, 61, 58, 56, 60, 76, 78, 71

Offset: 1

Clark Kimberling, Sep 09 2011

See A194959 for a discussion of fractalization and the interspersion fractally induced by a sequence.
The sequence A004736 is the fractal sequence obtained by concatenating the segments 1; 2,1; 3,2,1; 4,3,2,1;...
Every pair of rows of A195108 eventually intersperse.
As a sequence, A194108 is a permutation of the positive integers, with inverse A195109.

			Northwest corner:
1...2...5...8...13..19..26
3...6...10..15..21..28..36
4...7...11..17..23..30..39
9...14..20..27..35..44..54
12..18..24..32..41..51..62

Cf. A194959, A004736, A195107, A195109.

j[n_] := Table[n + 1 - k, {k, 1, n}]; t[1] = j[1];
t[n_] := Join[t[n - 1], j[n]]   (* A004736 *)
t[10]
p[n_] := t[20][[n]]
g[1] = {1}; g[n_] := Insert[g[n - 1], n, p[n]]
f[1] = g[1]; f[n_] := Join[f[n - 1], g[n]]
f[20] (* A195107 *)
row[n_] := Position[f[30], n];
u = TableForm[Table[row[n], {n, 1, 5}]]
v[n_, k_] := Part[row[n], k];
w = Flatten[Table[v[k, n - k + 1], {n, 1, 13},
{k, 1, n}]] (* A195108 *)
q[n_] := Position[w, n]; Flatten[Table[q[n],
{n, 1, 80}]] (* A195109 *)

A128139 Triangle read by rows: matrix product A004736 * A128132.

Original entry on oeis.org

1, 1, 2, 1, 3, 3, 1, 4, 5, 4, 1, 5, 7, 7, 5, 1, 6, 9, 10, 9, 6, 1, 7, 11, 13, 13, 11, 7, 1, 8, 13, 16, 17, 16, 13, 8, 1, 9, 15, 19, 21, 21, 19, 15, 9, 1, 10, 17, 22, 25, 26, 25, 22, 17, 10

Offset: 0

Gary W. Adamson, Feb 16 2007

A077028 with the final term in each row omitted.
Interchanging the factors in the matrix product leads to A128140 = A128132 * A004736.
From Gary W. Adamson, Jul 01 2012: (Start)
Alternatively, antidiagonals of an array A(n,k) of sequences with arithmetic progressions as follows:
  1, 2, 3,  4,  5,  6, ...
  1, 3, 5,  7,  9, 11, ...
  1, 4, 7, 10, 13, 16, ...
  1, 5, 9, 13, 17, 21, ...
  ... (End)
From Gary W. Adamson, Jul 02 2012: (Start)
A summation generalization for Sum_{k>=1} 1/(A(n,k)*A(n,k+1)) (formulas copied from A002378, A000466, A085001, A003185):
  1 = 1/(1)*(2) + 1/(2)*(3) + 1/(3)*(4)  + ...
  1 = 2/(1)*(3) + 2/(3)*(5) + 2/(5)*(7)  + ...
  1 = 3/(1)*(4) + 3/(4)*(7) + 3/(7)*(10) + ...
  1 = 4/(1)*(5) + 4/(5)*(9) + 4/(9)*(13) + ...
  ...
As a summation of terms equating to a definite integral:
  Integral_{0..1} dx/(1+x) = ... 1 - 1/2 + 1/3 - 1/4 + ... = log(2).
  Integral_{0..1} dx/(1+x^2) = 1 - 1/3 + 1/5 - 1/7  + ... = Pi/4 (see A157142)
  Integral_{0..1} dx/(1+x^3) = 1 - 1/4 + 1/7 - 1/10 + ... (see A016777)
  Integral_{0..1} dx/(1+x^4) = 1 - 1/5 + 1/9 - 1/13 + ... (see A016813). (End)

			First few rows of the triangle:
  1;
  1,  2;
  1,  3,  3;
  1,  4,  5,  4;
  1,  5,  7,  7,  5;
  1,  6,  9, 10,  9,  6;
  1,  7, 11, 13, 13, 11,  7;
  1,  8, 13, 16, 17, 16, 13,  8;
  1,  9, 15, 19, 21, 21, 19, 15,  9;
  1, 10, 17, 22, 25, 26, 25, 22, 17, 10;
  ...

Cf. A004736, A128132, A128140, A004006 (row sums).

A004736 * A128132 as infinite lower triangular matrices.

T(n,k) = k*(1+n-k)+1 = 1 + A094053(n+1,1+n-k). - R. J. Mathar, Jul 09 2012

A128225 A127899 (unsigned) * A004736.

Original entry on oeis.org

1, 6, 2, 15, 9, 3, 28, 20, 12, 4, 45, 35, 25, 15, 5, 66, 54, 42, 30, 18, 6, 91, 77, 63, 49, 35, 21, 7, 120, 104, 88, 72, 56, 40, 24, 8, 153, 135, 117, 99, 81, 63, 45, 27, 9, 190, 170, 150, 130, 110, 90, 70, 50, 30, 10

Offset: 1

Gary W. Adamson, Feb 19 2007

Row sums = the cubes, A000578: (1, 8, 27, 64, 125, ...). Left column = the hexagonal numbers: A000384: (1, 6, 15, 28, ...). A128226 = A004736 * A127899.

			First few rows of the triangle are:
   1;
   6,  2;
  15,  9,  3;
  28, 20, 12,  4;
  45, 35, 25, 15,  5;
  66, 54, 42, 30, 18,  6;
  91, 77, 63, 49, 35, 21,  7;
  ...

Cf. A000384, A000578, A004736, A099375, A127899, A128226.

(* a127899U computes the unsigned version of A127899 *)
a127899U[n_, k_] := If[n==k||n-1==k, n, 0]/;(1<=k<=n)
a004736[n_, k_] := n-k+1/;(1<=k<=n+1)
a128225[n_, k_] := a127899U[n, n](a004736[n, k] + a004736[n-1, k])/;(1<=k<=n)
a128225[r_] := Table[a128225[n, k], {n, 1, r}, {k, 1, n}]
TableForm[a128225[7]] (* triangle *)
Flatten[a128225[10]] (* data *) (* Hartmut F. W. Hoft, Mar 13 2017 *)

A127899 (unsigned) * A004736, as infinite lower triangular matrices. Triangle read by rows: n*[(1); (3,1); (5,3,1);...]; cf. A099375.

A135152 A004736 + A128174 - I, I = Identity matrix.

Original entry on oeis.org

1, 2, 1, 4, 2, 1, 4, 4, 2, 1, 6, 4, 4, 2, 1, 6, 6, 4, 4, 2, 1, 8, 6, 6, 4, 4, 2, 1, 8, 8, 6, 6, 4, 4, 2, 1, 10, 8, 8, 6, 6, 4, 4, 2, 1, 10, 10, 8, 8, 6, 6, 4, 4, 2, 1

Offset: 1

Gary W. Adamson, Nov 21 2007

Row sums = A047838: (1, 3, 7, 11, 17, 23, 31, 39, ...). The triangle is a companion to A135151.

			First few rows of the triangle:
  1;
  2, 1;
  4, 2, 1;
  4, 4, 2, 1;
  6, 4, 4, 2, 1;
  6, 6, 4, 4, 2, 1;
  8, 6, 6, 4, 4, 2, 1;
  ...

Cf. A135151, A047838, A004736, A128174.

A004736 + A128174 - I, where I = Identity matrix, A004736 = (1; 2,1; 3,2,1; ...) and A128174 = (1; 0,1; 1,0,1; 0,1,0,1; ...).

A144680 Triangle read by rows, lower half of an array formed by A004736 * A144328 (transform).

Original entry on oeis.org

1, 2, 3, 3, 5, 7, 4, 7, 11, 14, 5, 9, 15, 21, 25, 6, 11, 19, 28, 36, 41, 7, 13, 23, 35, 47, 57, 63, 8, 15, 27, 42, 58, 73, 85, 92, 9, 17, 31, 49, 69, 89, 107, 121, 129, 10, 19, 35, 56, 80, 105, 129, 150, 166, 175

Offset: 1

Gary W. Adamson, Sep 19 2008

Triangle read by rows, lower half of an array formed by A004736 * A144328 (transform).

			The array is formed by A004736 * A144328 (transform) where A004736 = the natural number decrescendo triangle and A144328 = a crescendo triangle. First few rows of the array =
  1, 1,  1,  1,  1,  1, ...
  2, 3,  3,  3,  3,  3, ...
  3, 5,  7,  7,  7,  7, ...
  4, 7, 11, 14, 14, 14, ...
  5, 9, 15, 21, 25, 25, ...
  ...
Triangle begins as:
   1;
   2,  3;
   3,  5,  7;
   4,  7, 11, 14;
   5,  9, 15, 21, 25;
   6, 11, 19, 28, 36,  41;
   7, 13, 23, 35, 47,  57,  63;
   8, 15, 27, 42, 58,  73,  85,  92;
   9, 17, 31, 49, 69,  89, 107, 121, 129;
  10, 19, 35, 56, 80, 105, 129, 150, 166, 175;
  ...

G. C. Greubel, Rows n = 1..50 of the triangle, flattened

Cf. A004006, A006008, A027965, A050407, A074742, A144328.

T[n_, k_]:= (3*(k^2-k+2)*n - k*(k-1)*(2*k-1))/6;
Table[T[n, k], {n,12}, {k,n}]//Flatten (* G. C. Greubel, Oct 18 2021 *)

def A144680(n,k): return (3*(k^2-k+2)*n - k*(k-1)*(2*k-1))/6
flatten([[A144680(n,k) for k in (1..n)] for n in (1..12)]) # G. C. Greubel, Oct 18 2021

Sum_{k=1..n} T(n, k) = A006008(n).
From G. C. Greubel, Oct 18 2021: (Start)
T(n, k) = (1/6)*( 3*(k^2 - k + 2)*n - k*(k-1)*(2*k-1) ).
T(n, n) = A004006(n).
T(n, n-1) = A050407(n+2).
T(n, n-2) = A027965(n-1) = A074742(n-2). (End)

cp's OEIS Frontend

A056534 Mapping from the ordering by product (A027750, A056538) to the ordering by sum (A002260, A004736) of ordered pairs (a,b), a>=1, b>=1.

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Maple

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A158823 Triangle read by rows: matrix product A004736 * A158821.

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Magma

Maple

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Sage

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A195107 Fractalization of the fractal sequence A004736. Interspersion fractally induced by A004736.

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A093919 Consider the triangle in A004736, the k-th term of the n-th row is the LCM of the preceding k terms.

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A173395 a(n) = (A002260(n) + 1) * (A004736(n) + 1).

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PARI

PARI

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A195108 Interspersion fractally induced by A004736.

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Mathematica

A128139 Triangle read by rows: matrix product A004736 * A128132.

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A128225 A127899 (unsigned) * A004736.

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