cp's OEIS Frontend

Also k is included if (and only if) the greatest power of 2 dividing k is >= the highest odd divisor of k. All terms of the sequence are even besides the 1.

Equivalently, positive integers of the form k*2^m, where odd k <= 2^m. - Thomas Ordowski, Oct 19 2014

If we define a divisor d|n to be superior if d >= n/d, then superior divisors are counted by A038548 and listed by A161908. This sequence consists of 1 and all numbers without a superior odd divisor. - Gus Wiseman, Feb 18 2021

Numbers k such that A006519(k) >= A000265(k), with equality only when k = 1. - Amiram Eldar, Jan 24 2023

Also number of partitions of n with even number of even parts. There is no restriction on the odd parts.

a(n) = u(n) + v(n), n >= 2, of the Osima reference, p. 383.

Also number of partitions of n with largest part congruent to n modulo 2: a(2*n) = A027187(2*n), a(2*n-1) = A027193(2*n-1); a(n) = A000041(n) - A000701(n). - Reinhard Zumkeller, Apr 22 2006

Equivalently, number of partitions of n with number of parts having the same parity as n. - Olivier Gérard, Apr 04 2012

Also number of distinct free Young diagrams (Ferrers graphs with n nodes). Free Young diagrams are distinct when none is a rigid transformation (translation, rotation, reflection or glide reflection) of another. - Jani Melik, May 08 2016

Let the cycle type of an even permutation be represented by the partition A=(O1,O2,...,Oi,E1,E2,...,E2j), where the Os are parts with odd length and the Es are parts with even lengths, and where j may be zero, using Reinhard Zumkeller's observation that the partition associated with a cycle type of an even permutation has an even number of even parts. The set of even cycle types enumerated here can be considered a monoid under a binary operation *: Let A be as above and B=(o1,o2,...,ok,e1,e2,...,e2m). A*B is the partition (O1o1,O1o2,...,O1ok,O1e1,...,O1e2m,O2o1,...,O2e2m,...,Oio1,...,Oie2m,E1o1,...,E1e2m,...,E2je2m). This product has 2im+2jk+4jm even parts, so it represents the cycle type of an even permutation. - Richard Locke Peterson, Aug 20 2018

From Gus Wiseman, Mar 31 2022: (Start)

Also the number of integer partitions of n with Heinz number greater than or equal to that of their conjugate, where the Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). These partitions are ranked by A352488. The complement is counted by A000701. For example, the a(n) partitions for n = 1...7 are:

(1) (11) (21) (22) (221) (222) (331)

(111) (211) (311) (321) (2221)

(1111) (2111) (2211) (3211)

(11111) (3111) (4111)

(21111) (22111)

(111111) (31111)

(211111)

(1111111)

Also the number of integer partitions of n with Heinz number less than or equal to their conjugate, ranked by A352489. For example, the a(n) partitions for n = 1...7 are:

(1) (2) (3) (4) (5) (6) (7)

(21) (22) (32) (33) (43)

(31) (41) (42) (52)

(311) (51) (61)

(321) (322)

(411) (421)

(511)

(4111)

(End)

a(n) = #{d : d = A182469(n,k), d <= A000196(n), k=1..A001227(n)}. - Reinhard Zumkeller, Apr 05 2015

The partitions of n are partitioned into four types:

EO, even # of odd parts and odd # of even parts, A236559;

OE, odd # of odd parts and even # of even parts, A160786;

EE, even # of odd parts and even # of even parts, A236913;

OO, odd # of odd parts and odd # of even parts, A236914.

A236559 and A160786 are the bisections of A027193;

A236913 and A236914 are the bisections of A027187.

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 its negated first-differences.

The reverse-alternating sum of a partition (y_1,...,y_k) is Sum_i (-1)^(k-i) y_i.

Also the number of reversed of integer partitions of n with alternating sum < 0.

No integer partitions have alternating sum < 0, so the non-reversed version is all zeros.

Is this sequence weakly increasing? Note: a(2n + 2) = A236914(n), a(2n) = A344743(n).

A formula for the reverse-alternating sum of a partition is: (-1)^(k-1) times the number of odd parts in the conjugate partition, where k is the number of parts. So a(n) is the number of integer partitions of n of even length whose conjugate parts are not all odd. Partitions of the latter type are counted by A086543. By conjugation, a(n) is also the number of integer partitions of n of even maximum whose parts are not all odd.

The partitions of n are partitioned into four types:

EO, even # of odd parts and odd # of even parts, A236559;

OE, odd # of odd parts and even # of even parts, A160786;

EE, even # of odd parts and even # of even parts, A236913;

OO, odd # of odd parts and odd # of even parts, A236914.

A236559 and A160786 are the bisections of A027193;

A236913 and A236914 are the bisections of A027187.