cp's OEIS Frontend

Found when generalizing Matula numbers to labeled trees.

The rows and columns are all increasing, and every prime occurs exactly once.

A007097(n) = Min {k : A109301(k) = n} = the first k whose rote height is n, the level set leader or minimum inverse function corresponding to A109301. - Jon Awbrey, Jun 26 2005

Lubomir Alexandrov informs me that he studied this sequence in his 1965 notebook. - N. J. A. Sloane, May 23 2008

a(n) is the Matula-Goebel number of the rooted path tree on n+1 vertices. The Matula-Goebel number of a rooted tree can be defined in the following recursive manner: to the one-vertex tree there corresponds the number 1; to a tree T with root degree 1 there corresponds the t-th prime number, where t is the Matula-Goebel number of the tree obtained from T by deleting the edge emanating from the root; to a tree T with root degree m>=2 there corresponds the product of the Matula-Goebel numbers of the m branches of T. - Emeric Deutsch, Feb 18 2012

Conjecture: log(a(1))*log(a(2))*...*log(a(n)) ~ a(n). - Thomas Ordowski, Mar 26 2015

Background discussion: Suppose that s is an increasing sequence of positive integers, that the complement t of s is infinite, and that t(1)=1. The dispersion of s is the array D whose n-th row is (t(n), s(t(n)), s(s(t(n))), s(s(s(t(n)))), ...). Every positive integer occurs exactly once in D, so that, as a sequence, D is a permutation of the positive integers. The sequence u given by u(n)=(number of the row of D that contains n) is a fractal sequence. Examples:

(1) s=A000040 (the primes), D=A114537, u=A114538.

(2) s=A022342 (without initial 0), D=A035513 (Wythoff array), u=A003603.

(3) s=A007067, D=A035506 (Stolarsky array), u=A133299.

More recent examples of dispersions: A191426-A191455.

The array is read by antidiagonals: A(1,1), A(1,2), A(2,1), A(1,3), A(2,2), A(3,1), etc.

This array can be obtained by taking every second column from array A242378, starting from its column 2.

Permutation of natural numbers larger than 1.

The terms on row n are all divisible by n-th prime, A000040(n).

Each column is strictly growing, and the terms in the same column have the same prime signature.

A055396(n) gives the row number of row where n occurs,

and A246277(n) gives its column number, both starting from 1.

From Antti Karttunen, Jan 03 2015: (Start)

A252759(n) gives their sum minus one, i.e. the Manhattan distance of n from the top left corner.

If we assume here that a(1) = 1 (but which is not explicitly included because outside of the array), then A252752 gives the inverse permutation. See also A246276.

(End)

A self-inverse permutation of the positive integers.

Background discussion: Suppose that s is an increasing sequence of positive integers, that the complement t of s is infinite, and that t(1)=1. The dispersion of s is the array D whose n-th row is (t(n), s(t(n)), s(s(t(n))), s(s(s(t(n)))), ...). Every positive integer occurs exactly once in D, so that, as a sequence, D is a permutation of the positive integers. The sequence u given by u(n)=(number of the row of D that contains n) is a fractal sequence. Examples:

(1) s=A000040 (the primes), D=A114537, u=A114538.

(2) s=A022343 (without initial 0), D=A035513 (Wythoff array), u=A003603.

(3) s=A007067, D=A035506 (Stolarsky array), u=A133299.

More recent examples of dispersions: A191426-A191455.

Fernandez calls this the order of primeness of n.

a(A007097(n))=n, for any n >= 0. - Paul Tek, Nov 12 2013

When a nonoriented rooted tree is encoded as a Matula-Goebel number n, a(n) tells how many edges needs to be climbed up from the root of the tree until the first branching vertex (or the top of the tree, if n is one of the terms of A007097) is encountered. Please see illustrations at A061773. - Antti Karttunen, Jan 27 2014

Zero-based column index of n in the Kimberling-style dispersion table of the primes (see A114537). - Allan C. Wechsler, Jan 09 2024

From Clark Kimberling, Mar 18 2015 and May 21 2015: (Start)

This is the signature sequence of log(3)/log(2) and is a fractal sequence; e.g., if the first occurrence of each n is removed, the resulting sequence is the original sequence.

Moreover, if the sequence is partitioned into segments starting with 0 as follows:

0,1

0,2,1,3

0,2,4,1,3

0,5,2,4,1,6,3,

and so on, then deleting the greatest number in each segment leaves

0

0,2,1

0,2,1,3

0,5,2,4,1,3,

and so on, which, concatenated to (0,0,2,1,0,2,1,3,0,5,2,4,1,3,...), is another fractal sequence, in today's usual meaning of that term. When introduced in 1995, one of the defining properties of a fractal sequence was, essentially, that before each n appears, every k < n must have already appeared; this requirement ensures that the sequence yields a dispersion; e.g., A114577 yields A114537. However, the usual meaning of "fractal sequence" nowadays is simply "a sequence that contains itself as a proper subsequence". It is proposed here that the original version be renamed "strongly fractal". Thus, the operations called upper trimming and lower trimming (e.g., A084531, A167237), when applied to strongly fractal sequences, yield strongly fractal sequences. The operation introduced here, which can be called "segment-upper trimming", carries fractal sequences to fractal sequences, but not strongly fractal to strongly fractal.

Associated with the signature sequence S of each positive irrational number is an interspersion (or equivalently, a dispersion), in which row n >= 0 consists of the positions of n in S. The interspersion associated with the signature sequence of log(3)/log(2) is A255975.

(End)

Comment from Allan C. Wechsler, May 26 2024 (Start):

More generally, the "signature sequence" of an irrational number H can be defined as follows. Consider all the numbers of the form a + bH, where a and b are positive integers, and sort them into increasing order (there are no cluster points or other obstacles). The sequence of a-values is then the *signature sequence of H.

If the coefficients a and b are allowed to be 0, you get the same sequence but with all the entries decremented by 1.

(End)

a(n) = A069352(n) - A022329(n). - Reinhard Zumkeller, May 16 2015

Suppose that s is an increasing sequence of positive integers, that the complement t of s is infinite, and that t(1)=1. The dispersion of s is the array D whose n-th row is (t(n), s(t(n)), s(s(t(n))), s(s(s(t(n)))), ...). Every positive integer occurs exactly once in D, so that, as a sequence, D is a permutation of the positive integers. The sequence u given by u(n) = {index of the row of D that contains n} is a fractal sequence. In this case s(n) = A016789(n-1), t(n) = A032766(n) [from term A032766(1) onward] and u(n) = A253887(n). [Author's original comment edited by Antti Karttunen, Jan 24 2015]

For other examples of such sequences, please see the Crossrefs section.