A208147 -id:A208147 - cp's OEIS frontend

A089080 Sequence is S(infinity) where S(1)={1,2} and S(n)=S(n-1)S'(n-1), where S'(k) is obtained from S(k) by replacing the single 1 with the least integer not occurring in S(k).

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

Offset: 1

Benoit Cloitre, Dec 04 2003

			S(1)={1,2} then S'(1)={3,2} and sequence begins 1,2,3,2

Antti Karttunen, Table of n, a(n) for n = 1..16384

Essentially the same as A085058 (with prepended 1 and different indexing).

Cf. A094267 (first differences), A208147 (partial products).

A085058(n) = (valuation(n+1, 2)+2);
A089080(n) = if(1==n,n,A085058(n-2)); \\ Antti Karttunen, Nov 01 2018

Sum_{k=1..n} a(k) = 3*n+O(log(n)) ( Sum_{k=1..n} a(k) < 3*n )

A356639 Number of integer sequences b with b(1) = 1, b(m) > 0 and b(m+1) - b(m) > 0, of length n which transform under the map S into a nonnegative integer sequence. The transform c = S(b) is defined by c(m) = Product_{k=1..m} b(k) / Product_{k=2..m} (b(k) - b(k-1)).

Original entry on oeis.org

1, 1, 3, 17, 155, 2677, 73327, 3578339, 329652351

Offset: 1

Thomas Scheuerle, Aug 19 2022

This sequence can be calculated by a recursive algorithm:
Let B1 be an array of finite length, the "1" denotes that it is the first generation. Let B1' be the reversed version of B1. Let C be the element-wise product C = B1 * B1'. Then B2 is a concatenation of taking each element of B1 and add all divisors of the corresponding element in C. If we start with B1 = {1} then we get this sequence of arrays: B2 = {2}, B3 = {3, 4, 6}, ... . a(n) is the length of the array Bn. In short the length of Bn+1 and so a(n+1) is the sum over A000005(Bn * Bn').
The transform used in the definition of this sequence is its own inverse, so if c = S(b) then b = S(c). The eigensequence is 2^n = S(2^n).
There exist some transformation pairs of infinite sequences in the database:
  A000027 <--> A000142; A000079 <--> A000079; A000045 <--> A001654;
  A026549 <--> A038754; A100071 <--> A001405; A058295 <--> A------;
  A111286 <--> A098011; A093968 <--> A205825; A166447 <--> A------;
  A098558 <--> A004277; A010551 <--> A087811; A100538 <--> A329227;
  A079352 <--> A------; A082458 <--> A------; A008233 <--> A264635;
  A138278 <--> A------; A006501 <--> A264557; A336496 <--> A------;
  A019464 <--> A------; A062112 <--> A------; A171647 <--> A359039;
  A279312 <--> A------; A031923 <--> A------.
These transformation pairs are conjectured:
  A137326 <--> A------; A066332 <--> A300902; A208147 <--> A308546;
  A057895 <--> A------; A349080 <--> A------; A019442 <--> A------;
  A349079 <--> A------.
("A------" means not yet in the database.)
Some sequences in the lists above may need offset adjustment to force a beginning with 1,2,... in the transformation.
If we allowed signed rational numbers, further interesting transformation pairs could be observed. For example, 1/n will transform into factorials with alternating sign. 2^(-n) transforms into ones with alternating sign and 1/A000045(n) into A000045 with alternating sign.

			a(4) = 17. The 17 transformation pairs of length 4 are:
  {1, 2, 3, 4}  = S({1, 2, 6, 24}).
  {1, 2, 3, 5}  = S({1, 2, 6, 15}).
  {1, 2, 3, 6}  = S({1, 2, 6, 12}).
  {1, 2, 3, 9}  = S({1, 2, 6, 9}).
  {1, 2, 3, 12} = S({1, 2, 6, 8}).
  {1, 2, 3, 21} = S({1, 2, 6, 7}).
  {1, 2, 4, 5}  = S({1, 2, 4, 20}).
  {1, 2, 4, 6}  = S({1, 2, 4, 12}).
  {1, 2, 4, 8}  = S({1, 2, 4, 8}).
  {1, 2, 4, 12} = S({1, 2, 4, 6}).
  {1, 2, 4, 20} = S({1, 2, 4, 5}).
  {1, 2, 6, 7}  = S({1, 2, 3, 21}).
  {1, 2, 6, 8}  = S({1, 2, 3, 12}).
  {1, 2, 6, 9}  = S({1, 2, 3, 9}).
  {1, 2, 6, 12} = S({1, 2, 3, 6}).
  {1, 2, 6, 15} = S({1, 2, 3, 5}).
  {1, 2, 6, 24} = S({1, 2, 3, 4}).
b(1) = 1 by definition, b(2) = 1+1 as 1 has only 1 as divisor.
a(3) = A000005(b(2)*b(2)) = 3.
The divisors of b(2) are 1,2,4. So b(3) can be b(2)+1, b(2)+2 and b(2)+4.
a(4) = A000005((b(2)+1)*(b(2)+4)) + A000005((b(2)+2)*(b(2)+2)) + A000005((b(2)+4)*(b(2)+1)) = 17.

Cf. A000005.
Cf. A000027, A000045, A000079, A000142, A001405.
Cf. A001654, A004277, A006501, A008233, A019442.
Cf. A010551, A019464, A026549, A031923, A038754.
Cf. A057895, A058295, A062112, A066332, A100071.
Cf. A079352, A082458, A087811, A093968, A098011.
Cf. A098558, A100538, A111286, A166447, A137326.
Cf. A138278, A171647, A205825, A208147, A264557.
Cf. A264635, A308546, A329227, A336496, A349079.
Cf. A349080, A359039.

A208152 Triangle by rows, generated from the Ruler sequence A001511.

Original entry on oeis.org

1, 1, 1, 3, 2, 1, 6, 4, 1, 1, 24, 16, 4, 3, 1, 48, 32, 8, 6, 1, 1, 144, 96, 24, 18, 3, 2, 1, 188, 192, 48, 36, 6, 4, 1, 1, 1440, 960, 240, 180, 30, 20, 5, 4, 1, 2880, 1920, 480, 360, 60, 10, 8, 1, 1, 8640, 5760, 1440, 1080, 180, 120, 30, 24, 3, 2, 1

Offset: 1

Gary W. Adamson, Feb 23 2012

Row sums = A208147: (1, 2, 6, 12, 48, 96, 288, ...).

			First few rows of the triangle =
     1;
     1,    1;
     3,    2,    1;
     6,    4,    1,    1;
    24,   16,    4,    3,   1;
    48,   32,    8,    6,   1,   1;
   144,   96,   24,   18,   3,   2,  1;
   288,  192,   48,   36,   6,   4,  1,  1;
  1440,  960,  240,  180,  30,  20,  5,  4, 1;
  2880, 1920,  480,  360,  60,  40, 10,  8, 1, 1;
  8640, 5760, 1440, 1080, 180, 120, 30, 24, 3, 2, 1;
  ...

Cf. A001511, A208147.

Inverse of:
   1;
  -1,  1;
  -1, -2,  1;
  -1, -2, -1,  1;
  -1, -2, -1, -3,  1;
  -1, -2, -1, -3, -1,  1;
  -1, -2, -1, -3, -1, -2,  1;
  ..., where the signed terms are negatives of A001511 terms: (1, 2, 1, 3, 1, 2, 1, 4, ...).

cp's OEIS Frontend

A089080 Sequence is S(infinity) where S(1)={1,2} and S(n)=S(n-1)S'(n-1), where S'(k) is obtained from S(k) by replacing the single 1 with the least integer not occurring in S(k).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

PARI

Formula

A356639 Number of integer sequences b with b(1) = 1, b(m) > 0 and b(m+1) - b(m) > 0, of length n which transform under the map S into a nonnegative integer sequence. The transform c = S(b) is defined by c(m) = Product_{k=1..m} b(k) / Product_{k=2..m} (b(k) - b(k-1)).

Views

Author

Keywords

Comments

Examples

Crossrefs

A208152 Triangle by rows, generated from the Ruler sequence A001511.

Views

Author

Keywords

Comments

Examples

Crossrefs

Formula