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The decapitation of such a partition (delete the greatest part) is term-wise less than its negated first-differences.

The decapitation of such a partition (delete the greatest part) is term-wise greater than or equal to its negated first-differences.

The decapitation of such a partition (delete the greatest part) is term-wise greater than its negated first-differences.

The decapitation of such a partition (delete the greatest part) is term-wise greater than or equal to its negated first-differences.

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)

Also Heinz numbers of integer partitions with no adjacent parts having quotient > 2 (counted by A342094). The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), giving a bijective correspondence between positive integers and integer partitions.

We define a divisor d|n to be strictly inferior if d < n/d. Strictly inferior divisors are counted by A056924 and listed by A341674.

These chains have first-quotients (in analogy with first-differences) that are term-wise > their decapitation (maximum element removed). Equivalently, x > y^2 for all adjacent x, y. For example, the divisor chain q = 60/6/2/1 has first-quotients (10,3,2), which are > (6,2,1), so q is counted under a(60).

Also the number of factorizations of n where each factor is greater than the product of all previous factors.

We define a divisor d|n to be strictly superior if d > n/d. Strictly superior divisors are counted by A056924 and listed by A341673.

These chains have first-quotients (in analogy with first-differences) that are term-wise < their decapitation (maximum element removed). Equivalently, x < y^2 for all adjacent x, y. For example, the divisor chain q = 30/6/3 has first-quotients (5,2), which are < (6,3), so q is counted under a(30).

Also the number of ordered factorizations of n where each factor is less than the product of all previous factors.

An alternative wording: Number of chains of divisors starting with n and having all adjacent parts x > y^2.

We define a divisor d|n to be superior if d >= n/d. Superior divisors are counted by A038548 and listed by A161908.

These chains have first-quotients (in analogy with first-differences) that are term-wise less than or equal to their decapitation (maximum element removed). Equivalently, x <= y^2 for all adjacent x, y. For example, the divisor chain q = 24/8/4/2 has first-quotients (3,2,2), which are less than or equal to (8,4,2), so q is counted under a(24).

Also the number of ordered factorizations of n where each factor is less than or equal to the product of all previous factors.