A002260 -id:A002260 - cp's OEIS frontend

A195110 Fractalization of the fractal sequence A002260. Interspersion fractally induced by A002260.

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

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 A002260 is the fractal sequence obtained by concatenating the segments 1; 12; 123; 1234; 12345;...

Cf. A194959, A002260, A195111, A195112.

j[n_] := Table[k, {k, 1, n}]; t[1] = j[1];
t[n_] := Join[t[n - 1], j[n]]   (* A002260 *)
t[12]
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] (* A195110 *)
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}]] (* A195111 *)
q[n_] := Position[w, n]; Flatten[Table[q[n], {n, 1, 80}]]  (* A195112 *)

A195111 Interspersion fractally induced by the fractal sequence A002260.

Original entry on oeis.org

1, 3, 2, 6, 4, 5, 10, 8, 9, 7, 15, 13, 14, 11, 12, 21, 19, 20, 16, 17, 18, 28, 26, 27, 23, 24, 25, 22, 36, 34, 35, 31, 32, 33, 29, 30, 45, 43, 44, 40, 41, 42, 37, 38, 39, 55, 53, 54, 50, 51, 52, 46, 47, 48, 49, 66, 64, 65, 61, 62, 63, 57, 58, 59, 60, 56, 78, 76, 77

Offset: 1

Clark Kimberling, Sep 09 2011

See A194959 for a discussion of fractalization and the interspersion fractally induced by a sequence.
Every pair of rows eventually intersperse.  As a sequence, A194111 is a permutation of the positive integers, with inverse A195129.
The sequence A002260 is the fractal sequence obtained by concatenating the segments 1; 12; 123; 1234; 12345;...

			Northwest corner:
1...3...6...10..15..21..28..36..45
2...4...8...13..19..26..34..43..53
5...9...14..20..27..35..44..54..65
7...11..16..23..31..40..50..61..73
12..17..24..32..41..51..62..74..87

Cf. A194959, A002260, A195110, A195112.

j[n_] := Table[k, {k, 1, n}]; t[1] = j[1];
t[n_] := Join[t[n - 1], j[n]]   (* A002260 *)
t[12]
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] (* A195110 *)
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}]] (* A195111 *)
q[n_] := Position[w, n]; Flatten[Table[q[n],
  {n, 1, 80}]]  (* A195112 *)

A220280 The reluctant sequence for A002260.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 1, 2, 1, 2, 3, 1, 1, 2, 1, 2, 3, 1, 1, 1, 2, 1, 2, 3, 1, 2, 1, 1, 2, 1, 2, 3, 1, 2, 3, 1, 1, 2, 1, 2, 3, 1, 2, 3, 4, 1, 1, 2, 1, 2, 3, 1, 2, 3, 4, 1, 1, 1, 2, 1, 2, 3, 1, 2, 3, 4, 1, 2, 1, 1, 2, 1, 2, 3, 1, 2, 3, 4, 1, 2, 3

Offset: 1

Boris Putievskiy, Dec 12 2012

The reluctant sequence B for a sequence A is a triangular array in which row k (>= 1) consists of the first k terms of A.

Here A002260 is the reluctant sequence for the sequence 1,2,3,... of positive numbers (A000027).

			A002260 begins
  1, 1, 2, 1, 2, 3, 1, 2, 3, 4, 1, 2, 3, 4, 5, 1, 2, 3, 4, 5, 6, 1, 2, 3, 4, 5, 6, 7, ...
so the first few rows of the new triangle are
   1,
   1, 1,
   1, 1, 2,
   1, 1, 2, 1,
   1, 1, 2, 1, 2,
   1, 1, 2, 1, 2, 3,
   ...
                                                                               ~

Boris Putievskiy, Table of n, a(n) for n = 1..10000
Boris Putievskiy, Transformations Integer Sequences And Pairing Functions, arXiv:1212.2732 [math.CO], 2012.

Cf. A002260, A002262, A037126, A059268, A215026.

t=int((math.sqrt(8*n-7) - 1)/ 2)
n1=n-t*(t+1)/2
t1=int((math.sqrt(8*n1-7) - 1)/ 2)
a=n1-t1*(t1+1)/2

a(n) = n1 - t1(t1+1)/2, where n1 = n - t(t+1)/2, t1 = floor[(-1+sqrt(8*n1-7))/2], t=floor[(-1+sqrt(8*n-7))/2]. For example, a(6)=2 since t=2, t1=1, n1=3.

Edited by N. J. A. Sloane, Jun 07 2024

A101387 Quet transform of A002260.

Original entry on oeis.org

1, 1, 2, 1, 3, 1, 5, 1, 1, 7, 1, 2, 1, 9, 1, 3, 1, 12, 1, 1, 4, 1, 15, 1, 2, 1, 5, 1, 18, 1, 3, 1, 7, 1, 1, 21, 1, 4, 1, 9, 1, 2, 1, 24, 1, 5, 1, 11, 1, 3, 1, 28, 1, 1, 6, 1, 13, 1, 4, 1, 32, 1, 2, 1, 7, 1, 15, 1, 5, 1, 36, 1, 3, 1, 9, 1, 1, 17, 1, 6, 1, 40, 1, 4, 1, 11, 1, 2, 1, 19, 1, 7, 1, 44, 1, 5, 1

Offset: 1

David Wasserman, Jan 14 2005

The Quet transform converts any sequence of positive integers containing an infinite number of 1's into another sequence of positive integers containing an infinite number of 1's.
Start with a sequence, {a(k)}, of only positive integers and an infinite number of 1's. Example: 1,1,2,1,2,3,1,2,3,4,1,... (A002260).
Form the sequence {b(k)} (which is a permutation of the positive integers), given by b(k) = the a(k)th smallest positive integer not yet in the sequence b, with b(1)=a(1).
In the example b is 1,2,4,3,6,8,5,9,11,13,7,12,15,... (A065562).
Let {c(k)} be the inverse of {b(k)}. In the example c = 1,2,4,3,7,5,11,6,8,16,9,12... (A065579).
Form the final sequence {d(k)}, where each d(k) is such that c(k) = the d(k)th smallest positive integer not yet in the sequence c, with d(1)=c(1).
In the example d is 1,1,2,1,3,1,5,1,1,7,1,2,1,9,1,3,1,12,1,1,4,1,15,... (the current sequence).
A more formal description of the Quet transform is as follows.
Let N denote the positive integers. For any permutation p: N -> N, let T(p): N -> N be given by T(p)(n) = # of elements in {m in N | m >= n AND p(m) <= p(n)}. Observe that T is a bijection from the set of permutations N -> N onto the set of sequences N -> N that contain infinitely many 1's.
Now suppose f: N -> N contains infinitely many 1's; then its Quet transform Q(f): N -> N is T^(-1)[(T(f))^(-1)], which also contains infinitely many 1's. Q is self-inverse; f and Q(f) correspond via T to a permutation and its inverse.

Lars Blomberg, Table of n, a(n) for n = 1..10000
David Wasserman, Quet transform + PARI code [Cached copy]

Cf. A002260, A100661.

\\ PARI code to compute the Quet transform.  Put the first n terms of the sequence
\\ into a vector v; then Q(v) returns the transformed sequence.  The output is a
\\ vector, containing as many terms as can be computed from the given data.
TInverse(v) = local(l, w, used, start, x); l = length(v); w = vector(l); used = vector(l); start = 1; for (i = 1, l, while (start <= l && used[start], start++); x = start; for (j = 2, v[i], x++; while (x <= l && used[x], x++)); if (x > l, return (vector(i - 1, k, w[k])), w[i] = x; used[x] = 1)); w;
PInverse(v) = local(l, w); l = length(v); w = vector(l); for (i = 1, l, if (v[i] <= l, w[v[i]] = i)); w;
T(v) = local(l, w, c); l = length(v); w = vector(l); for (n = 1, l, if (v[n], c = 0; for (m = 1, n - 1, if (v[m] < v[n], c++)); w[n] = v[n] - c, return (vector(n - 1, i, w[i])))); w;
Q(v) = T(PInverse(TInverse(v)));
\\ David Wasserman, Jan 14 2005

A128077 A128064 * A002260.

Original entry on oeis.org

1, 1, 4, 1, 2, 9, 1, 2, 3, 16, 1, 2, 3, 4, 25, 1, 2, 3, 4, 5, 36, 1, 2, 3, 4, 5, 6, 49, 1, 2, 3, 4, 5, 6, 7, 64

Offset: 1

Gary W. Adamson, Feb 14 2007

Row sums = the pentagonal numbers, A000326: 1, 5, 12, 22, 35, 51, ...

			First few rows of the triangle:
  1;
  1, 4;
  1, 2, 9;
  1, 2, 3, 16;
  1, 2, 3,  4, 25;
  ...

Cf. A128064, A002260, A000326, A128078.

A128064 * A002260 as infinite lower triangular matrices. Triangle read by rows, a(1) = 1; n-th row = first (n-1) terms of (1, 2, 3, ...) followed by n^2.

A128180 A002260 * A097807.

Original entry on oeis.org

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

Offset: 1

Gary W. Adamson, Feb 17 2007

Row sums = A008794: (1, 1, 4, 4, 9, 9, 16, 16, ...).
Unsigned row sums = the triangular sequence, A000217: (1, 3, 6, 10, ...) by virtue of the fact that each row is a permutation of the natural numbers.
A128179 = A097807 * A002260.

			Triangle begins:
   1;
  -1,  2;
   2, -1,  3;
  -2,  3, -1,  4;
   3, -2,  4, -1,  5;
  -3,  4, -2,  5, -1,  6;
   4, -3,  5, -2,  6, -1,  7;
  ...

Cf. A002260, A097807, A008794, A128179.

T(n,k)=(2*k-1+(-1)^(n-k)*(2*n+1))/4 \\ Franklin T. Adams-Watters, Apr 12 2011

A002260 * A097807 as infinite lower triangular matrices.
From Franklin T. Adams-Watters, Apr 12 2011: (Start)
T(n,k) = (2k - 1 + (-1)^(n-k)*(2n+1))/4.
|T(n,k)| = (2n+1 + (-1)^(n-k)*(2k-1))/4. (End)

A128228 A128229 * A002260.

Original entry on oeis.org

1, 2, 2, 3, 6, 3, 4, 8, 12, 4, 5, 10, 15, 20, 5, 6, 12, 18, 24, 30, 6, 7, 14, 21, 28, 35, 42, 7, 8, 16, 24, 32, 40, 48, 56, 8, 9, 18, 27, 36, 45, 54, 63, 72, 9, 10, 20, 30, 40, 50, 60, 70, 80, 90, 10

Offset: 1

Gary W. Adamson, Feb 19 2007

Row sums = A006000: (1, 4, 12, 28, 55, 96, 154,...).

			First few rows of the triangle are:
1;
2, 2;
3, 6, 3;
4, 8, 12, 4;
5, 10, 15, 20, 5;
6, 12, 18, 24, 30, 6;
7, 14, 21, 28, 35, 42, 7;
...

Cf. A128229, A002260, A006000, A128227.

(* first n rows of the triangle *)
a128228[n_] := Table[If[r==q, r, q r], {r, 1, n}, {q, 1, r}]
Flatten[a128228[10]] (* data *)
TableForm[a128228[7]] (* triangle *)
(* Hartmut F. W. Hoft, Jun 10 2017 *)

A128229 * A002260 as infinite lower triangular matrices.
Triangle, n * (each term of A128227).
T(n,k) = k*n if 1<=kHartmut F. W. Hoft, Jun 10 2017

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).

A195113 Fractalization of the fractal sequence obtained by deleting the second two terms of the fractal sequence A002260.

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Sep 09 2011

nonn

See A194959 for a discussion of fractalization and the interspersion fractally induced by a sequence p; for the present case, p is the concatenation of the segments 1, 123,1234,12345,123456,..., so that p is obtained from A002260 by deleting the segment 12.

Cf. A194959, A002260, A195114, A195115.

j[n_] := Table[k, {k, 1, n}];
t[1] = j[1]; t[2] = j[1];
t[n_] := Join[t[n - 1], j[n]] (* A002260; initial 1,1,2 repl by 1 *)
t[12]
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]  (* A195113 *)
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}]] (* A195114 *)
q[n_] := Position[w, n]; Flatten[Table[q[n],
{n, 1, 80}]]  (* A195115 *)

A195114 Interspersion fractally induced by the fractal sequence obtained by deleting the second two terms of the fractal sequence A002260.

Original entry on oeis.org

1, 3, 2, 6, 4, 5, 10, 7, 8, 9, 15, 12, 13, 14, 11, 21, 18, 19, 20, 16, 17, 28, 25, 26, 27, 22, 23, 24, 36, 33, 34, 35, 29, 30, 31, 32, 45, 42, 43, 44, 38, 39, 40, 41, 37, 55, 52, 53, 54, 48, 49, 50, 51, 46, 47, 66, 63, 64, 65, 59, 60, 61, 62, 56, 57, 58, 78, 75, 76

Offset: 1

Clark Kimberling, Sep 09 2011

See A194959 for a discussion of fractalization and the interspersion fractally induced by a sequence. Every pair of rows eventually intersperse. As a sequence, A194114 is a permutation of the positive integers, with inverse A195115.

			Northwest corner:
1...3...6...10..15..21..28
2...4...7...12..18..25..33
5...8...13..19..26..34..43
9...14..20..27..35..44..54
11..16..22..29..38..48..59

Cf. A194959, A002260, A195113, A195115, A195111.

j[n_] := Table[k, {k, 1, n}];
t[1] = j[1]; t[2] = j[1];
t[n_] := Join[t[n - 1], j[n]] (* A002260; initial 1,1,2 repl by 1 *)
t[12]
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]  (* A195113 *)
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}]] (* A195114 *)
q[n_] := Position[w, n]; Flatten[Table[q[n],
{n, 1, 80}]]  (* A195115 *)

cp's OEIS Frontend

A195110 Fractalization of the fractal sequence A002260. Interspersion fractally induced by A002260.

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

A195111 Interspersion fractally induced by the fractal sequence A002260.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A220280 The reluctant sequence for A002260.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Python

Formula

Extensions

A101387 Quet transform of A002260.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

A128077 A128064 * A002260.

Views

Author

Keywords

Comments

Examples

Crossrefs

Formula

A128180 A002260 * A097807.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

PARI

Formula

A128228 A128229 * A002260.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Formula