cp's OEIS Frontend

Numbers n such that sum of divisors of n (A000203) is odd.

Also the numbers with an odd number of run sums (trapezoidal arrangements, number of ways of being written as the difference of two triangular numbers). - Ron Knott, Jan 27 2003

Pell(n)*Sum_{k|n} 1/Pell(k) is odd, where Pell(n) is A000129(n). - Paul Barry, Oct 12 2005

Number of odd divisors of n (A001227) is odd. - Vladeta Jovovic, Aug 28 2007

A071324(a(n)) is odd. - Reinhard Zumkeller, Jul 03 2008

Sigma(a(n)) = A000203(a(n)) = A152677(n). - Jaroslav Krizek, Oct 06 2009

Numbers n such that sum of odd divisors of n (A000593) is odd. - Omar E. Pol, Jul 05 2016

A187793(a(n)) is odd. - Timothy L. Tiffin, Jul 18 2016

If k is odd (k = 2m+1 for m >= 0), then 2^k = 2^(2m+1) = 2*(2^m)^2. If k is even (k = 2m for m >= 0), then 2^k = 2^(2m) = (2^m)^2. So, the powers of 2 sequence (A000079) is a subsequence of this one. - Timothy L. Tiffin, Jul 18 2016

Numbers n such that A175317(n) = Sum_{d|n} pod(d) is odd, where pod(m) = the product of divisors of m (A007955). - Jaroslav Krizek, Dec 28 2016

Positions of zeros in A292377 and A292383, positions of ones in A286357 and A292583. (See A292583 for why.) - Antti Karttunen, Sep 25 2017

Numbers of the form A000079(i)*A016754(j), i,j>=0. - R. J. Mathar, May 30 2020

Equivalently, numbers whose odd part is square. Cf. A042968. - Peter Munn, Jul 14 2020

These are the Heinz numbers of the partitions counted by A119620. - Gus Wiseman, Oct 29 2021

Numbers m whose abundance, A033880(m), is odd. - Peter Munn, May 23 2022

Numbers with an odd number of middle divisors (cf. A067742). - Omar E. Pol, Aug 02 2022

If n = prime(x_1) * prime(x_2) * prime(x_3) * ... * prime(x_k) then a(n) = x_1 - x_2 + x_3 - ... + (-1)^(k-1) x_k, where the x_i are weakly increasing positive integers.

The value of a(n) depends only on the squarefree part of n, A007913(n). - Antti Karttunen, May 06 2022

The alternating sum of a partition (y_1,...,y_k) is Sum_i (-1)^(i-1) y_i, which is equal to the number of odd parts in the conjugate partition.

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.

Also the reverse-alternating sum of the prime indices of n.

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

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

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 whose conjugate parts are all even or whose length is odd. By conjugation, this is also the number of integer partitions of n whose parts are all even or whose greatest part is odd.

All integer partitions have alternating sum >= 0, so the non-reversed version is A000041.

Is this sequence weakly increasing? In particular, is A344611(n) <= A160786(n)?

Interleaving of A114121 and A032443. Row sums of A116405. Binomial transform is A116409.

Appears to be the number of n-digit binary numbers not having more zeros than ones; equivalently, the number of unrestricted Dyck paths of length n not going below the axis. - Ralf Stephan, Mar 25 2008

From Gus Wiseman, Jun 20 2021: (Start)

Also the number compositions of n with alternating sum >= 0, where the alternating sum of a sequence (y_1,...,y_k) is Sum_i (-1)^(i-1) y_i. The a(0) = 1 through a(5) = 11 compositions are:

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

(11) (21) (22) (32)

(111) (31) (41)

(112) (113)

(121) (122)

(211) (212)

(1111) (221)

(311)

(1121)

(2111)

(11111)

(End)

From J. Stauduhar, Jan 14 2022: (Start)

Also, for n >= 2, first differences of partial row sums of Pascal's triangle. The first ceiling(n/2)+1 elements of rows n=0 to n=4 in Pascal's triangle are:

1

1 1

1 2

1 3 3

1 4 6

...

The cumulative sums of these partial rows form the sequence 1,3,6,13,24,..., and its first differences are a(2),a(3),a(4),... in this sequence.

(End)

The alternating sum of a partition (y_1,...,y_k) is Sum_i (-1)^(i-1) y_i. This is equal to the number of odd parts in the conjugate partition, so T(n,k) is the number of integer partitions of n with k odd parts in the conjugate partition, which is also the number of partitions of n with k odd parts.

Also the number of integer partitions of n with odd-indexed parts (odd bisection) summing to k, ceiling(n/2) <= k <= n. The even-indexed version is A346633. - Gus Wiseman, Nov 29 2021

Up to sign, same as A124754.

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

The k-th composition in standard order (graded reverse-lexicographic, A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again. This gives a bijective correspondence between nonnegative integers and integer compositions.

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

The k-th composition in standard order (graded reverse-lexicographic, A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again. This gives a bijective correspondence between nonnegative integers and integer compositions.

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

Also the number of reversed integer partitions of 2n with alternating sum >= 0.

The reverse-alternating sum of a partition is equal to (-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 partitions of 2n whose conjugate parts are all even or whose length is odd. By conjugation, this is also the number of partitions of 2n whose parts are all even or whose greatest part is odd.

From Gus Wiseman, Jun 08 2021: (Start)

Also the number of:

- integer partitions of 2n with reverse-alternating sum 2;

- reversed integer partitions of 2n with alternating sum 2;

- integer partitions of 2n with exactly two odd parts, one of which is the greatest;

- odd-length integer partitions of 2n whose conjugate partition has exactly two odd parts.

Note that integer partitions of 2n with alternating or reverse-alternating sum 0 are counted by A000041, ranked by A000290.

(End)