cp's OEIS Frontend

Meyers (see Guy reference) conjectures that for all r >= 1, the least odd number not in the set {a(i): i < prime(r)} is prime(r+1). - N. J. A. Sloane, Jan 08 2021

Meyers' conjecture would be refuted if and only if for some r there were such a large gap between prime(r) and prime(r+1) that there existed a composite c for which prime(r) < c < a(c) < prime(r+1), in which case (by Bertrand's postulate) c would necessarily be a term of A246281. - Antti Karttunen, Mar 29 2021

a(n) is odd for all n and for each odd m there exists a k with a(k) = m (see A064216). a(n) > n for n > 1: bijection between the odd and all numbers. - Reinhard Zumkeller, Sep 26 2001

a(n) and n have the same number of distinct primes with (A001222) and without multiplicity (A001221). - Michel Marcus, Jun 13 2014

From Antti Karttunen, Nov 01 2019: (Start)

More generally, a(n) has the same prime signature as n, A046523(a(n)) = A046523(n). Also A246277(a(n)) = A246277(n) and A287170(a(n)) = A287170(n).

Many permutations and other sequences that employ prime factorization of n to encode either polynomials, partitions (via Heinz numbers) or multisets in general can be easily defined by using this sequence as one of their constituent functions. See the last line in the Crossrefs section for examples.

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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).

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

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

A001221(a(n)) = A000120(n) for any n >= 0.

From Antti Karttunen, Jan 01 2019: (Start)

A034684(a(n)) = A000961(1+A001511(n)) for any n >= 1. (See also Rémy Sigrist's comment in A289271).

This sequence can be regarded also as an irregular triangle with rows of lengths 1, 1, 2, 4, 8, 16, ..., that is, it can be represented as a binary tree, where each left hand child contains A322991(k), and each right hand child contains A322992(k), when their parent contains k:

1

|

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

3 6

4......../ \........10 12......../ \........30

/ \ / \ / \ / \

/ \ / \ / \ / \

/ \ / \ / \ / \

5 14 15 42 20 70 60 210

7 18 21 66 28 90 84 330 35 126 105 462 140 630 420 2310

etc.

The leftmost edge is A000961, the next lefmost is A278568 (after 2: 6, 10, 14, 18, ...), the righmost edge is A002110, the next rightmost A088860 but with 3 instead of 4.

Compare also to trees like A005940 (A163511) and A052330.

(End)

For all n > 1, in the binary tree illustrated in A289272, the node which contains (has value) n, its parent node has value a(n).

Each n occurs exactly twice in this sequence.