cp's OEIS Frontend

The binary representation of a(n) shows which prime numbers divide n, but not the multiplicities. a(2)=1, a(3)=10, a(4)=1, a(5)=100, a(6)=11, a(10)=101, a(30)=111, etc.

For n > 1, a(n) gives the (one-based) index of the column where n is located in array A285321. A008479 gives the other index. - Antti Karttunen, Apr 17 2017

From Antti Karttunen, Jun 18 & 20 2017: (Start)

A268335 gives all n such that a(n) = A248663(n); the squarefree numbers (A005117) are all the n such that a(n) = A285330(n) = A048675(n).

For all n > 1 for which the value of A285331(n) is well-defined, we have A285331(a(n)) <= floor(A285331(n)/2), because then n is included in the binary tree A285332 and a(n) is one of its ancestors (in that tree), and thus must be at least one step nearer to its root than n itself.

Conjecture: Starting at any n and iterating the map n -> a(n), we will always reach 0 (see A288569). This conjecture is equivalent to the conjecture that at any n that is neither a prime nor a power of two, we will eventually hit a prime number (which then becomes a power of two in the next iteration). If this conjecture is false then sequence A285332 cannot be a permutation of natural numbers. On the other hand, if the conjecture is true, then A285332 must be a permutation of natural numbers, because all primes and powers of 2 occur in definite positions in that tree. This conjecture also implies the conjectures made in A019565 and A285320 that essentially claim that there are neither finite nor infinite cycles in A019565.

If there are any 2-cycles in this sequence, then both terms of the cycle should be present in A286611 and the larger one should be present in A286612.

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Binary rank of the distinct prime indices of n, where the binary rank of an integer partition y is given by Sum_i 2^(y_i-1). For all prime indices (with multiplicity) we have A048675. - Gus Wiseman, May 25 2024

An integer partition is double-free if no part is twice any other part.

The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).

An integer partition is double-free if no part is twice any other part.

Also numbers n such that if prime(m) divides n then prime(2m) does not divide n, i.e., numbers not divisible by any element of A319613.

Complement of A158582. - Reinhard Zumkeller, Apr 16 2009

Also union of A168604 and A030130. - Douglas Latimer, Jul 19 2012

Numbers of the form 2^t - 2^k - 1, 0 <= k < t.

n is in the sequence if and only if 2*n+1 is in the sequence. - Robert Israel, Dec 14 2018

Also the least binary rank of a strict integer partition of n, where the binary rank of a partition y is given by Sum_i 2^(y_i-1). - Gus Wiseman, May 24 2024

See A010060, the main entry for the Thue-Morse sequence, for additional information. - N. J. A. Sloane, Aug 13 2014

a(A000069(n)) = -1; a(A001969(n)) = +1. - Reinhard Zumkeller, Apr 29 2012

Partial sums of every third terms give A005599. - Reinhard Zumkeller, May 26 2013

Fixed point of the morphism 1 --> 1,-1 and -1 --> -1,1. - Robert G. Wilson v, Apr 07 2014

Fibbinary numbers (A003714) gives the numbers n for which a(n) = A132971(n). - Antti Karttunen, May 30 2017

Also partitions whose multiplicities are all 2's, except possibly for the last, which may be 1.

The number of binary strings sharing the same autocorrelations.

Appears to be row sums of A155092, beginning from a(2). - Mats Granvik, Jan 20 2009

The number of binary words of length n (beginning with 0) which do not start with an even palindrome (i.e. which are not of the form ss*t where s is a (nonempty) word, s* is its reverse, and t is any (possibly empty) word). - Mamuka Jibladze, Sep 30 2014

From Gus Wiseman, Mar 08 2021: (Start)

This sequence counts each of the following essentially equivalent things:

1. Sets of distinct positive integers with maximum n in which all adjacent elements have quotients > 1/2. For example, the a(1) = 1 through a(6) = 10 sets are:

{1} {2} {3} {4} {5} {6}

{2,3} {3,4} {3,5} {4,6}

{2,3,4} {4,5} {5,6}

{2,3,5} {3,4,6}

{3,4,5} {3,5,6}

{2,3,4,5} {4,5,6}

{2,3,4,6}

{2,3,5,6}

{3,4,5,6}

{2,3,4,5,6}

2. For n > 1, sets of distinct positive integers with maximum n - 1 whose first-differences are term-wise less than their decapitation (remove the maximum). For example, the set q = {2,4,5} has first-differences (2,1), which are not less than (2,4), so q is not counted under a(5). On the other hand, r = {2,3,5,6} has first-differences {1,2,1}, which are less than {2,3,5}, so r is counted under a(6).

3. Compositions of n where each part after the first is less than the sum of all preceding parts. For example, the a(1) = 1 through a(6) = 10 compositions are:

(1) (2) (3) (4) (5) (6)

(21) (31) (41) (51)

(211) (32) (42)

(311) (411)

(212) (321)

(2111) (312)

(3111)

(2121)

(2112)

(21111)

(End)

Apparently a(n) = number of partitions (p_1, p_2, ..., p_k) of n+1, with p_1 >= p_2 >= ... >= p_k, such that for each i, p_i > p_{i+1}+...+p_k. - John McKay (mac(AT)mathstat.concordia.ca), Mar 06 2009

Comment from John McKay confirmed in paper by Bessenrodt, Olsson, and Sellers. Such partitions are called "strongly decreasing" partitions in the paper, see the function s(n) therein.

Also the number of unlabeled binary rooted trees with 2*n + 3 nodes in which the two branches directly under any given non-leaf node are either equal or at least one of them is a leaf. - Gus Wiseman, Oct 08 2018

From Gus Wiseman, Apr 06 2021: (Start)

This sequence counts both of the following essentially equivalent things:

1. Sets of distinct positive integers with maximum n + 1 in which all adjacent elements have quotients < 1/2. For example, the a(0) = 1 through a(8) = 7 subsets are:

{1} {2} {3} {4} {5} {6} {7} {8} {9}

{1,3} {1,4} {1,5} {1,6} {1,7} {1,8} {1,9}

{2,5} {2,6} {2,7} {2,8} {2,9}

{3,7} {3,8} {3,9}

{1,3,7} {1,3,8} {4,9}

{1,3,9}

{1,4,9}

2. Sets of distinct positive integers with maximum n + 1 whose first differences are term-wise greater than their decapitation (remove the maximum). For example, the set q = {1,4,9} has first differences (3,5), which are greater than (1,4), so q is counted under a(8). On the other hand, r = {1,5,9} has first differences (4,4), which are not greater than (1,5), so r is not counted under a(8).

Also the number of partitions of n + 1 into powers of 2 covering an initial interval of powers of 2. For example, the a(0) = 1 through a(8) = 7 partitions are:

1 11 21 211 221 2211 421 4211 4221

111 1111 2111 21111 2221 22211 22221

11111 111111 22111 221111 42111

211111 2111111 222111

1111111 11111111 2211111

21111111

111111111

(End)