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This is a permutation of the positive integers.

a(1) could be taken as 0 because 1 is not a member of A001414 and one could start with a(0)=1 (see the W. Lang link).

The row length sequence of this array is A000607(n), n>=2.

If the array is [1,0,2,3,4,5,6,6,...] with offset 0 then the row length sequence is A000607(n), n>=0.

From David James Sycamore, May 11 2018: (Start)

For n > 1, a(n) is the smallest number not yet seen such that sopfr(a(n)) is the least possible integer. The sequence lists in increasing order elements of the finite sets S(k) = {x: sopfr(x)=k}, k >= 0, where sopfr(x) = 0 iff x = 1. When a(n) = A056240(k) for some k >= 2, then sopfr(a(n)) = k and a(n) is the first of A000607(k) terms, all of which have sopfr = k. (A000607(k) is the number of partitions of k into prime parts.) Consequently the sequence follows a sawtooth profile, rising from a(n) = A056240(k) to A000792(k), the greatest number with sopfr = k, then starting over with A056240(k+1) for the next larger value of sopfr. (End) [Edited by M. F. Hasler, Jan 19 2019]

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

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=0 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. For large n, the sequence is recursively defined.

After a(7), this appears to be the same as A319054.

This sequence is log-superadditive, i.e., a(m+n) >= a(m)*a(n). By Fekete's subadditive lemma, it follows that the limit of a(n)^(1/n) exists and equals the supremum of a(n)^(1/n). - Pontus von Brömssen, Mar 03 2022

a(10) >= 105.

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 =4 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 5 and 6.]

3. The function a(n) = maximum value of the product of the terms k_i, where Sum (k_i) = n+ 4 - 4*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=0 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 = 3.

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

5. For large n, the sequence is recursively defined.

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=0 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 = 3.

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