cp's OEIS Frontend

Restricted growth sequence transform of the ordered pair [A003557(n), A329345(n)].

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.

(End)

Prime product form of primorial base expansion of n.

Sequence is a permutation of A048103. It maps the smallest prime not dividing n to the smallest prime dividing n, that is, A020639(a(n)) = A053669(n) holds for all n >= 1.

The sequence satisfies the exponential function identity, a(x + y) = a(x) * a(y), whenever A329041(x,y) = 1, that is, when adding x and y together will not generate any carries in the primorial base. Examples of such pairs of x and y are A328841(n) & A328842(n), and also A328770(n) (when added with itself). - Antti Karttunen, Oct 31 2019

From Antti Karttunen, Feb 18 2022: (Start)

The conjecture given in A327969 asks whether applying this function together with the arithmetic derivative (A003415) in some combination or another can eventually transform every positive integer into zero.

Another related open question asks whether there are any other numbers than n=6 such that when starting from that n and by iterating with A003415, one eventually reaches a(n). See comments in A351088.

This sequence is used in A351255 to list the terms of A099308 in a different order, by the increasing exponents of the successive primes in their prime factorization. (End)

From Bill McEachen, Oct 15 2022: (Start)

From inspection, the least significant decimal digits of a(n) terms form continuous chains of 30 as follows. For n == i (mod 30), i=0..5, there are 6 ordered elements of these 8 {1,2,3,6,9,8,7,4}. Then for n == i (mod 30), i=6..29, there are 12 repeated pairs = {5,0}.

Moreover, when the individual elements of any of the possible groups of 6 are transformed via (7*digit) (mod 10), the result matches one of the other 7 groupings (not all 7 may be seen). As example, {1,2,3,6,9,8} transforms to {7,4,1,2,3,6}. (End)

The least significant digit of a(n) in base 4 is given by A353486, and in base 6 by A358840. - Antti Karttunen, Oct 25 2022, Feb 17 2024

From Antti Karttunen, May 12 2014: (Start)

a(A003961(n)) = n for all n. [This is a left inverse function for the injection A003961.]

Bisections are A064216 (the terms at odd indices) and A064989 itself (the terms at even indices), i.e., a(2n) = a(n) for all n.

(End)

From Antti Karttunen, Dec 18-21 2014: (Start)

When n represents an unordered integer partition via the indices of primes present in its prime factorization (for n >= 2, n corresponds to the partition given as the n-th row of A112798) this operation subtracts one from each part. If n is of the form 2^k (a partition having just k 1's as its parts) the result is an empty partition (which is encoded by 1, having an "empty" factorization).

For all odd numbers n >= 3, a(n) tells which number is located immediately above n in square array A246278. Cf. also A246277.

(End)

Alternatively, if numbers are represented as the multiset of indices of prime factors with multiplicity, this operation subtracts 1 from each element and discards the 0's. - M. F. Hasler, Dec 29 2014

Grundy numbers of the game in which you decrease n by a number prime to n, and the game ends when 1 is reached. - Eric M. Schmidt, Jul 21 2013

a(n) = the smallest part of the partition having Heinz number n. We define the Heinz number of a partition p = [p_1, p_2, ..., p_r] 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] we get 2*2*3*7*29 = 2436. Example: a(21) = 2; indeed, the partition having Heinz number 21 = 3*7 is [2,4]. - Emeric Deutsch, Jun 04 2015

a(n) is the number of numbers whose largest proper divisor is n, i.e., for n>1, number of occurrences of n in A032742. - Stanislav Sykora, Nov 04 2016

For n > 1, a(n) gives the number of row where n occurs in arrays A083221 and A246278. - Antti Karttunen, Mar 07 2017

It seems that a(n) = Max_{j=n+1..2n-1} gcd(n,j). - Labos Elemer, May 22 2002

This is correct: No integer in the range [n+1, 2n-1] has n as its divisor, but certainly at least one multiple of the largest proper divisor of n will occur there (e.g., if it is n/2, then at n + (n/2)). - Antti Karttunen, Dec 18 2014

The slopes of the visible lines made by the points in the scatter plot are 1/2, 1/3, 1/5, 1/7, ... (reciprocals of primes). - Moosa Nasir, Jun 19 2022

From Antti Karttunen, Dec 06 2014: (Start)

For n >= 2, a(n) tells in which column of the sieve of Eratosthenes (see A083140, A083221) n occurs in. A055396 gives the corresponding row index.

(End)

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)

For any node n >= 2 in binary trees A005940 and A163511, a(n) gives the parent node of n. (Here we assume that their initial root 1 is its own parent).