cp's OEIS Frontend

From Peter Munn, Apr 06 2021: (Start)

a(n) is determined by the prime signature of n.

Compare with the multiplicative, self-inverse A225546, which also maps 2^e to the squarefree number A019565(e). However, this sequence maps p^e to the same squarefree number for every prime p, whereas A225546 maps the e-th power of progressively larger primes to progressively greater powers of A019565(e).

Both sequences map powers of squarefree numbers to powers of squarefree numbers.

(End)

Permutation of nonnegative integers. Note the indexing, the domain starts from 1, although the range includes also 0.

A246353 gives the inverse of this sequence, in a sense that a(A246353(n)) = n for all n >= 0, and A246353(a(n)) = n for all n >= 1. When one is subtracted from the latter, another permutation of nonnegative integers is obtained: A064273. - Antti Karttunen, Aug 23 2014 based on comment from Howard A. Landman, Sep 25 2001

Also index of n-th term of A019565 when its terms are sorted in increasing order. For example: a(6) = 8. The smallest values of A019565 are 1,2,3,5,6,7 . The 6th is 7 which is A019565(8). - Philippe Lallouet (philip.lallouet(AT)orange.fr), Apr 28 2008

a(n) is the number whose binary indices are the prime indices of the n-th squarefree number (row n of A329631), where a binary index of n is any position of a 1 in its reversed binary expansion, and a prime index of n is a number m such that prime(m) divides n. The binary indices of n are row n of A048793, while the prime indices of n are row n of A112798. - Gus Wiseman, Nov 30 2019

Note the indexing: the domain starts from 0, while the range excludes zero.

This sequence can be represented as a binary tree. Each left hand child is produced as A019565(n), and each right hand child as A065642(n), when the parent node contains n >= 2:

1

|

...................2...................

3 4

6......../ \........9 5......../ \........8

/ \ / \ / \ / \

/ \ / \ / \ / \

/ \ / \ / \ / \

15 12 14 27 10 25 7 16

210 45 35 18 105 28 462 81 21 20 154 125 30 49 11 32

etc.

Where will 38 appear in this tree? It is a reasonable assumption that by iterating A087207 starting from 38, as A087207(38) = 129, A087207(129) = 8194, A087207(8194) = 1501199875790187, ..., we will eventually hit a prime A000040(k), most likely with a largish index k. This prime occurs at the penultimate edge at right, as a(A000918(k)) = a((2^k)-2), and thus 38 occurs somewhere below it as a(m) = 38, m > k. All the numbers that share prime factors with 38, namely 76, 152, 304, 608, 722, ..., occur similarly late in this tree, as they form the rightward branch starting from 38. Alternatively, by iterating A285330 (each iteration moves one step towards the root) starting from 38, we might instead first hit some power of 3, or say, one of the terms of A033845 (the rightward branch starting from 6), in which case the first prime encountered would be a(2)=3 and 38 would appear on the left-hand side instead of the right-hand side subtree.

As long as it remains conjecture that A019565 has no cycles, it is certainly also an open question whether this is a permutation of the natural numbers: If A019565 has any cycles, then neither any of the terms in those cycles nor any A065642-trajectories starting from those terms (that is, numbers sharing same prime factors) may occur in this tree.

Sequence exhibits some outrageous swings, for example, a(703) = 224, but a(704) is 1427 decimal digits (4739 binary digits) long, thus it no longer fits into a b-file.

However, the scatter plot of A286543 gives some flavor of the behavior of this sequence even after that point. - Antti Karttunen, Dec 25 2017

This sequence is a variant of A256739; both sequences have nice graphical features.

Replacing "SumXOR" by "Sum" in the name leads to the Jordan function J_2 (A007434).

For any sequence f of nonnegative integers with positive indices:

- let x_f be the unique sequence satisfying SumXOR_{d divides n} x_f(d) = f(n) for any n > 0,

- in particular, x_A000027 = A256739 and x_A000290 = a (this sequence),

- also, x_A178910 = A000027 and x_A055895 = A000079,

- see the links section for a gallery of x_f plots for some classic f functions,

- x_f(1) = f(1),

- x_f(p) = f(1) XOR f(p) for any prime p,

- x_A063524 = A008966,

- x_f(n) = SumXOR_{d divides n and n/d is squarefree} f(d) for any n > 0,

- the function x: f -> x_f is a bijection,

- A000004 is the only fixed point of x (i.e. x_f = f if and only if f = A000004),

- for any sequence f, x_{2*f} = 2 * x_f,

- for any sequences g and f, x_{g XOR f} = x_g XOR x_f.

From Antti Karttunen, Dec 29 2017: (Start)

The transform x_f described above could be called "Xor-Moebius transform of f" because of its analogous construction to Möbius transform with A008683 replaced by A008966 and the summation replaced by cumulative XOR.

(End)

Each row beginning with an odd number (rows with even index) is followed by a row of the same length, with the same terms, but multiplied by 2. See also comments in the Formula section of A018819.

Note that although the indexing of rows start from zero, the indexing of this sequence starts from 1, with a(1) = 1.

Also Heinz numbers of integer partitions whose binary rank is n, where the binary rank of a partition y is given by Sum_i 2^(y_i-1). For example, row n = 6 is 15, 20, 27, 36, 48, 64, corresponding to the partitions (3,2), (3,1,1), (2,2,2), (2,2,1,1), (2,1,1,1,1), (1,1,1,1,1,1). - Gus Wiseman, May 25 2024

Also, row n lists in ascending order all A018819(n) numbers k for which A097248(k) = A019565(n). - Flávio V. Fernandes, Jul 19 2025

A permutation of the natural numbers > 1.

Otherwise like array A284311, but the columns come in different order.

For n > 0, with prime factorization Product_{i=1..k} p_i ^ e_i (all p_i distinct and all e_i > 0):

- let S_n = A000961 \ { p_i ^ (e_i + j) with i=1..k and j > 0 },

- a(n) = Sum_{i=1..k} 2^#{ s in S_n with 1 < s < p_i ^ e_i }.

In an informal way, we encode the prime powers > 1 that are unitary divisors of n as 1's in binary, while discarding the 0's corresponding to their "proper" multiples.

a(A002110(n)) = 2^n-1 for any n >= 0.

a(A000961(n+1)) = 2^(n-1) for any n > 0.

A000120(a(n)) = A001221(n) for any n > 0 (each prime divisor p of n (alongside the p-adic valuation of n) is encoded as a single 1 bit in the base-2 representation of a(n)).

A000961(2+A007814(a(n))) = A034684(n) for any n > 1 (the least significant bit of a(n) encodes the smallest unitary divisor of n that is larger than 1).

This sequence establishes a bijection between the positive numbers and the nonnegative numbers; see A289272 for the inverse of this sequence.

The numbers 4, 36, 40 and 532 equal their image; are there other such numbers?

This sequence has connections with A034729 (which encodes the divisors of a number, and is not surjective) and A087207 (which encodes the prime divisors of a number, and is not injective).