cp's OEIS Frontend

Numbers of the form 3^k, 2*3^k, 4*3^k with a(0) = 1 prepended.

If a set of positive numbers has sum n, this is the largest value of their product.

In other words, maximum of products of partitions of n: maximal value of Product k_i for any way of writing n = Sum k_i. To find the answer, take as many of the k_i's as possible to be 3 and then use one or two 2's (see formula lines below).

a(n) is also the maximal size of an Abelian subgroup of the symmetric group S_n. For example, when n = 6, one of the Abelian subgroups with maximal size is the subgroup generated by (123) and (456), which has order 9. [Bercov and Moser] - Ahmed Fares (ahmedfares(AT)my-deja.com), Apr 19 2001

Also the maximum number of maximal cliques possible in a graph with n vertices (cf. Capobianco and Molluzzo). - Felix Goldberg (felixg(AT)tx.technion.ac.il), Jul 15 2001 [Corrected by Jim Nastos and Tanya Khovanova, Mar 11 2009]

Every triple of alternate terms {3*k, 3*k+2, 3*k+4} in the sequence forms a geometric progression with first term 3^k and common ratio 2. - Lekraj Beedassy, Mar 28 2002

For n > 4, a(n) is the least multiple m of 3 not divisible by 8 for which omega(m) <= 2 and sopfr(m) = n. - Lekraj Beedassy, Apr 24 2003

Maximal number of divisors that are possible among numbers m such that A080256(m) = n. - Lekraj Beedassy, Oct 13 2003

Or, numbers of the form 2^p*3^q with p <= 2, q >= 0 and 2p + 3q = n. Largest number obtained using only the operations +,* and () on the parts 1 and 2 of any partition of n into these two summands where the former exceeds the latter. - Lekraj Beedassy, Jan 07 2005

a(n) is the largest number of complexity n in the sense of A005520 (A005245). - David W. Wilson, Oct 03 2005

a(n) corresponds also to the ultimate occurrence of n in A001414 and thus stands for the highest number m such that sopfr(m) = n, for n >= 2. - Lekraj Beedassy, Apr 29 2002

a(n) for n >= 1 is a paradigm shift sequence with procedural length p = 0, in the sense of A193455. - Jonathan T. Rowell, Jul 26 2011

a(n) = largest term of n-th row in A212721. - Reinhard Zumkeller, Jun 14 2012

For n >= 2, a(n) is the largest number whose prime divisors (with multiplicity) add to n, whereas the smallest such number (resp. smallest composite number) is A056240(n) (resp. A288814(n)). - David James Sycamore, Nov 23 2017

For n >= 3, a(n+1) = a(n)*(1 + 1/s), where s is the smallest prime factor of a(n). - David James Sycamore, Apr 10 2018

It is assumed that we start with a single letter in the copy buffer.

Alternatively, a(n-1) = maximal value of Product (k_i-1) for any way of writing n = Sum k_i.

1. The description above assumes that the text is deselected after the Copy command is invoked.

2. This sequence is the solution to the equivalent problem formulated as {insert, "Select All+ Copy" macro (without deselection), Paste}.

3. This sequence is a "paradigm-shift" sequence with procedure length p =1 (in the sense of A193455).

4. The optimal number of pastes per copy, as measured by the geometric growth rate (p+z root of z), is z = 4. [noninteger maximum between 3 and 4]

5. The function a(n) = maximum value of the product of the terms k_i, where Sum (k_i) = n+1-i_max.

6. All solutions will be of the form a(n) = m^b * (m+1)^d.

A "keystroke" means one of the following:

a (i.e., hit the letter "a" on the keyboard)

Ctrl-a ("select all")

Ctrl-c (copy selected text to clipboard)

Ctrl-v (paste from clipboard to cursor location)

Alternatively, a(n-2) = maximal value of Product (k_i-2) for any way of writing n = Sum k_i

1. Note that the copy command does not deselect the text.

2. This sequence is a "paradigm-shift" sequence with procedure length p =2 (in the sense of A193455).

3. The optimal number of pastes per copy, as measured by the geometric growth rate (p+z root of z), is z = 4. [Non-integer maximum between 4 and 5.]

4. The function a(n) = maximum value of the product of the terms k_i, where Sum (k_i) = n + 2 - 2*i_max.

5. All solutions will be of the form a(n) = m^b * (m+1)^d.

This sequence is the solution to the following problem: "Suppose you have the choice of using one of three production options: apply a simple incremental action, bundle existing output as an integrated product (which requires p=4 steps), or implement the current bundled action (which requires q=5 steps). The first use of a novel bundle erases (or makes obsolete) all prior actions. How large an output can be generated in n time steps?"

1. A production scheme with replacement R(p,q) eliminates existing output following a bundling action, while an additive scheme A(p,q) retains the output. The schemes correspond according to A(p,q)=R(p-q,q), with the replacement scheme serving as the default presentation.

2. This problem is structurally similar to the Copy and Paste Keyboard problem: Existing sequences (A178715 and A193286) should be regarded as Paradigm-Shift Sequences with production schemes R(1,1) and R(2,1) with replacement, respectively.

3. The ideal number of implementations per bundle, as measured by the geometric growth rate (p+zq root of z), is z = 3.

4. All solutions will be of the form a(n) = (qm+r) * m^b * (m+1)^d.

5. The optimal action string that generates this integer sequence (x=a^k PV^m ...) has inversions in the minimum number of bundling procedures P. These inversions are persistent as they are present when the recursive formula for the output first holds.

This sequence is the solution to the following problem: "Suppose you have the choice of using one of three production options: apply a simple action, bundle all existing actions (which requires p=3 steps), or apply the current bundled action. The first use of a novel bundle erases (or makes obsolete) all prior actions. How many total actions (simple) can be applied in n time steps?"

1. This problem is structurally similar to the Copy and Paste Keyboard problem: Existing sequences (A178715 and A193286) should be regarded as Paradigm-Shift Sequences with Procedural Lengths p=1 and 2, respectively.

2. The optimal number of pastes per copy, as measured by the geometric growth rate (p+z root of z), is z = 5. [Non-integer maximum between 4 and 5.]

3. The function a(n) = maximum value of the product of the terms k_i, where Sum (k_i) = n+ 3 - 3*i_max

4. All solutions will be of the form a(n) = m^b * (m+1)^d

This sequence is the solution to the following problem: "Suppose you have the choice of using one of three production options: apply a simple action, bundle all existing actions (which requires p=5 steps), or apply the current bundled action. The first use of a novel bundle erases (or makes obsolete) all prior actions. How many total actions (simple) can be applied in n time steps?"

1. This problem is structurally similar to the Copy and Paste Keyboard problem: Existing sequences (A178715 and A193286) should be regarded as Paradigm-Shift Sequences with Procedural Lengths p=1 and 2, respectively.

2. The optimal number of pastes per copy, as measured by the geometric growth rate (p+z root of z), is z = 6. [Non-integer maximum between 6 and 7.]

3. The function a(n) = maximum value of the product of the terms k_i, where Sum (k_i) = n+ 5 - 5*i_max

4. All solutions will be of the form a(n) = m^b * (m+1)^d

This sequence is the solution to the following problem: "Suppose you have the choice of using one of three production options: apply a simple incremental action, bundle existing output as an integrated product (which requires p=4 steps), or implement the current bundled action (which requires q=4 steps). The first use of a novel bundle erases (or makes obsolete) all prior actions. How large an output can be generated in n time steps?"

1. A production scheme with replacement R(p,q) eliminates existing output following a bundling action, while an additive scheme A(p,q) retains the output. The schemes correspond according to A(p,q)=R(p-q,q), with the replacement scheme serving as the default presentation.

2. This problem is structurally similar to the Copy and Paste Keyboard problem: Existing sequences (A178715 and A193286) should be regarded as Paradigm-Shift Sequences with production schemes R(1,1) and R(2,1) with replacement, respectively.

3. The ideal number of implementations per bundle, as measured by the geometric growth rate (p+zq root of z), is z = 4.

4. All solutions will be of the form a(n) = (qm+r) * m^b * (m+1)^d.

This sequence is the solution to the following problem: "Suppose you have the choice of using one of three production options: apply a simple incremental action, bundle existing output as an integrated product (which requires p=4 steps), or implement the current bundled action (which requires q=2 steps). The first use of a novel bundle erases (or makes obsolete) all prior actions. How large an output can be generated in n time steps?"

1. A production scheme with replacement R(p,q) eliminates existing output following a bundling action, while an additive scheme A(p,q) retains the output. The schemes correspond according to A(p,q)=R(p-q,q), with the replacement scheme serving as the default presentation.

2. This problem is structurally similar to the Copy and Paste Keyboard problem: Existing sequences (A178715 and A193286) should be regarded as Paradigm-Shift Sequences with production schemes R(1,1) and R(2,1) with replacement, respectively.

3. The ideal number of implementations per bundle, as measured by the geometric growth rate (p+zq root of z), is z = 4.

4. All solutions will be of the form a(n) = (qm+r) * m^b * (m+1)^d.

This sequence is the solution to the following problem: "Suppose you have the choice of using one of three production options: apply a simple incremental action, bundle existing output as an integrated product (which requires p=4 steps), or implement the current bundled action (which requires q=3 steps). The first use of a novel bundle erases (or makes obsolete) all prior actions. How large an output can be generated in n time steps?"

1. A production scheme with replacement R(p,q) eliminates existing output following a bundling action, while an additive scheme A(p,q) retains the output. The schemes correspond according to A(p,q)=R(p-q,q), with the replacement scheme serving as the default presentation.

2. This problem is structurally similar to the Copy and Paste Keyboard problem: Existing sequences (A178715 and A193286) should be regarded as Paradigm-Shift Sequences with production schemes R(1,1) and R(2,1) with replacement, respectively.

3. The ideal number of implementations per bundle, as measured by the geometric growth rate (p+zq root of z), is z = 4.

4. All solutions will be of the form a(n) = (qm+r) * m^b * (m+1)^d.