cp's OEIS Frontend

a(n) appears on row 1 of the array illustrated in A066099.

Except for initial zero, ordinal transform of A062050. After initial zero, n-th chunk consists of n, one n-1, two (n-2)'s, ..., 2^(k-1) (n-k)'s, ..., 2^(n-1) 1's. - Franklin T. Adams-Watters, Sep 11 2006

Zero together with a triangle read by rows in which row j lists the first 2^(j-1) terms of A001511 in nonincreasing order, j >= 1, see example. Also row j lists the first parts, in nonincreasing order, of the compositions of j. - Omar E. Pol, Sep 11 2013

The n-th row represents the frequency distribution of 1, 2, 3, ..., 2^(n-1) in the first 2^n - 1 terms of A003602. - Gary W. Adamson, Jun 10 2021

The representation of the compositions (for fixed n) is as lists of parts, the order between individual compositions (for the same n) is (list-)reversed lexicographic; see the example by Omar E. Pol. - Joerg Arndt, Sep 03 2013

This is the standard ordering for compositions in this database; it is similar to the Mathematica ordering for partitions (A080577). Other composition orderings include A124734 (similar to the Abramowitz & Stegun ordering for partitions, A036036), A108244 (similar to the Maple partition ordering, A080576), etc (see crossrefs).

Factorize each term in A057335; sequence records the values of the resulting exponents. It also runs through all possible permutations of multiset digits.

This can be regarded as a table in two ways: with each composition as a row, or with the compositions of each integer as a row. The first way has A000120 as row lengths and A070939 as row sums; the second has A001792 as row lengths and A001788 as row sums. - Franklin T. Adams-Watters, Nov 06 2006

This sequence includes every finite sequence of positive integers. - Franklin T. Adams-Watters, Nov 06 2006

Compositions (or ordered partitions) are also generated in sequence A101211. - Alford Arnold, Dec 12 2006

The equivalent sequence for partitions is A228531. - Omar E. Pol, Sep 03 2013

The sole partition of zero has no components, not a single component of length one. Hence the first nonempty row is row 1. - Franklin T. Adams-Watters, Apr 02 2014 [Edited by Andrey Zabolotskiy, May 19 2018]

See sequence A261300 for another version where the terms of each composition are concatenated to form one single integer: (0, 1, 2, 11, 3, 21, 12, 111,...). This also shows how the terms can be obtained from the binary numbers A007088, cf. Arnold's first Example. - M. F. Hasler, Aug 29 2015

The k-th composition in the list is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again. This is described as the standard ordering used in the OEIS, although the sister sequence A228351 is also sometimes considered to be canonical. Both sequences define a bijective correspondence between nonnegative integers and integer compositions. - Gus Wiseman, May 19 2020

First differences of A030303 = positions of bits 1 in the concatenation A030190 (= A030302) of numbers written in binary (A007088). - Indices of record values (= first occurrence of n) are given by A005183: a(A005183(n)) = n, cf. FORMULA for more. - M. F. Hasler, Oct 12 2020

The geometric mean approaches the Somos constant (A112302). - Jwalin Bhatt, Feb 10 2025

A permutation of the squarefree numbers A005117. The missing positive numbers are in A013929. - Alois P. Heinz, Sep 06 2014

From Antti Karttunen, Apr 18 & 19 2017: (Start)

Because a(n) toggles the parity of n there are neither fixed points nor any cycles of odd length.

Conjecture: there are no finite cycles of any length. My grounds for this conjecture: any finite cycle in this sequence, if such cycles exist at all, must have at least one member that occurs somewhere in A285319, the terms that seem already to be quite rare. Moreover, any such a number n should satisfy in addition to A019565(n) < n also that A048675^{k}(n) is squarefree, not just for k=0, 1 but for all k >= 0. As there is on average a probability of only 6/(Pi^2) = 0.6079... that any further term encountered on the trajectory of A048675 is squarefree, the total chance that all of them would be squarefree (which is required from the elements of A019565-cycles) is soon minuscule, especially as A048675 is not very tightly bounded (many trajectories seem to skyrocket, at least initially). I am also assuming that usually there is no significant correlation between the binary expansions of n and A048675(n) (apart from their least significant bits), or, for that matter, between their prime factorizations.

See also the slightly stronger conjecture in A285320, which implies that there would neither be any two-way infinite cycles.

If either of the conjectures is false (there are cycles), then certainly neither sequence A285332 nor its inverse A285331 can be a permutation of natural numbers. (End)

The conjecture made in A087207 (see also A288569) implies the two conjectures mentioned above. A further constraint for cycles is that in any A019565-trajectory which starts from a squarefree number (A005117), every other term is of the form 4k+2, while every other term is of the form 6k+3. - Antti Karttunen, Jun 18 2017

The sequence satisfies the exponential function identity, a(x + y) = a(x) * a(y), whenever x and y do not have a 1-bit in the same position, i.e., when A004198(x,y) = 0. See also A283475. - Antti Karttunen, Oct 31 2019

The above identity becomes unconditional if binary exclusive OR, A003987(.,.), is substituted for addition, and A059897(.,.), a multiplicative equivalent of A003987, is substituted for multiplication. This gives us a(A003987(x,y)) = A059897(a(x), a(y)). - Peter Munn, Nov 18 2019

Also the Heinz number of the binary indices of n, where the Heinz number of a sequence (y_1,...,y_k) is prime(y_1)*...*prime(y_k), and a number's binary indices (A048793) are the positions of 1's in its reversed binary expansion. - Gus Wiseman, Dec 28 2022

a(n) is also the sorted version of A057335 which is generated recursively using the formula A057335 = A057334 * A057335(repeated), where A057334 = A000040(A000120). - Alford Arnold, Nov 11 2001

Squarefree kernels of these numbers are primorial numbers. See A080404. - Labos Elemer, Mar 19 2003

If u and v are terms then so is u*v. - Reinhard Zumkeller, Nov 24 2004

Except for the initial value a(1) = 1, a(n) gives the canonical primal code of the n-th finite sequence of positive integers, where n = (prime_1)^c_1 * ... * (prime_k)^c_k is the code for the finite sequence c_1, ..., c_k. See examples of primal codes at A106177. - Jon Awbrey, Jun 22 2005

From Daniel Forgues, Jan 24 2011: (Start)

Least integer, in increasing order, of each ordered prime signature.

The least integer of each ordered prime signature are the smallest numbers with a given tuple of exponents of prime factors.

The ordered prime signature (where the order of exponents matters) of n corresponds to a given composition of Omega(n), as opposed to the prime signature of n, which corresponds to a given partition of Omega(n). (End)

Except for the initial entry 1, the entries of the sequence are the Heinz numbers of all partitions that contain all parts 1,2,...,k, where k is the largest part. The Heinz number of a partition p = [p_1, p_2, ..., p_r] is defined as Product(p_j-th prime, j=1...r) (concept used by Alois P. Heinz in A215366 as an "encoding" of a partition). For example, for the partition [1,1,2,4,10] the Heinz number is 2*2*3*7*29 = 2436. The number 150 (= 2*3*5*5) is in the sequence because it is the Heinz number of the partition [1,2,3,3]. - Emeric Deutsch, May 22 2015

Numbers n such that A053669(n) > A006530(n). - Anthony Browne, Jun 06 2016

From David W. Wilson, Dec 28 2018: (Start)

Numbers n such that for primes p > q, p | n => q | n.

Numbers n such that prime p | n => A034386(p) | n. (End)

Irregular triangle T(n,k) = first differences of indices of prime divisors p of n.

Row lengths = (big) Omega(n) = A001222(n).

Row sums = A061395(n).

Row maxima = A286469(n).

We can concatenate the rows 1 <= n <= 28 as none of the values of k in this range exceed 9: {0, 1, 2, 10, 3, 11, 4, 100, 20, 12, 5, 101, 6, 13, 21, 1000, 7, 110, 8, 102, 22, 14, 9, 1001, 30, 15, 200, 103}; a(29) = {10}, which would require a digit greater than 9.

a(1) = 0 by convention.

a(0) is not defined (i.e., null set). a(n) is defined for positive nonzero n.

a(p) = A000720(p) for p prime.

a(p^e) = A000720(p) followed by (e - 1) zeros.

a(Product(p^e)) is the concatenation of the a(p^e) of the unitary prime power divisors p^e of n, sorted by the prime p (i.e. the function a(n) mapped across the terms of row n of A141809).

a(A002110(n)) = an array of n 1s.

T(n,k) could be used to furnish A054841(n). We read data in row n of T(n,k). If T(n,1) = 0, then write 0. If T(n,1) > 0, then increment the k-th place from the right. For k > 1, increment the k-th place to the right of the last-incremented place.

T(n,k) can be used to render n in decimal. If T(n,1) = 0, then write 1. If T(n,1) > 0, then multiply 1 by A000720(T(n,1)). For k > 1, multiply the previous product by pi(x) = A000720(x) of the running total of T(n,k) for each k.

Ignoring zeros in row n > 1 and decoding the remaining values of T(n,k) as immediately above yields the squarefree kernel of n = A007947(n).

Leading zeros of a(n) are trimmed, but as in decimal notation numbers that include leading zeros symbolize the same n as without them. Zeros that precede nonzero values merely multiply implicit 1 by itself until we encounter nonzero values. Thus, {0,0,2} = 1*1*pi(2) = 3, as {2} = pi(2) = 3. Because of this no row n > 1 has 0 for k = 1 of T(n,k).

Interpreting n written in binary as a row of a(n) yields A057335(n).

Row k is the unique multiset that covers an initial interval of positive integers and has multiplicities equal to the parts of the k-th composition in standard order (graded reverse-lexicographic, A066099). This composition is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again. For example, the 13th composition is (1,2,1), so row 13 is {1,2,2,3}. - Gus Wiseman, Apr 26 2020

a(n) is the product of elements of the multiset that covers an initial interval of positive integers with multiplicities equal to the parts of the n-th composition in standard order (graded reverse-lexicographic, A066099). This composition is obtained by taking the set of positions of 1's in the reversed binary expansion of n, prepending 0, taking first differences, and reversing again. For example, the 13th composition is (1,2,1) giving the multiset {1,2,2,3} with product 12, so a(13) = 12. - Gus Wiseman, Apr 26 2020

For each of the A000041(n) partitions of n, one can assign a weight to the partition which counts the permutations of that partition, given by the multinomial coefficient derived from the frequency representation of the parts.

An equivalent representation is given by writing down all compositions of n.

The entries count those partitions multiplied by their weights (=compositions) of n where the sum of the largest addend plus number of parts equals k+1. Only nonzero counts are entered into the sequence.

Each entry can also be interpreted as counting a subset of numbers in A055932, because there is a 1-to-1 correspondence between their prime signature and ordered partitions.

Each diagonal of T(n,k) can be decomposed into p(n) sequences. For example,

A086602 = 2 12 39 95 195 ... is the sum of

A000330 = 1 5 14 30 55 ... plus

A001296 = 1 7 25 65 140 ...

The main diagonal and subdiagonals in order of appearance are A000124, A000330, A086602, A089574, A107600, A107601, A109125, ...

All terms sorted give A055932.

All terms first sorted by number of factors give A057335.