cp's OEIS Frontend

A way of writing n as a (nonnegative) linear combination of a finite sequence y is any sequence of pairs (k_i,y_i) such that k_i >= 0 and Sum k_i*y_i = n. For example, the pairs ((3,1),(1,1),(1,1),(0,2)) are a way of writing 5 as a linear combination of (1,1,1,2), namely 5 = 3*1 + 1*1 + 1*1 + 0*2. Of course, there are A000041(n) ways to write n as a linear combination of (1..n).

As a triangle, also the number of ways to write n as a *positive* linear combination of the parts of a strict integer partition of k.

From Gus Wiseman, Aug 28 2023: (Start)

Appears to be the number of subsets of {1..n} containing n such that no element can be written as a nonnegative linear combination of the others, first differences of A326083. For example, the a(1) = 1 through a(8) = 10 subsets are:

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

{2,3} {3,4} {2,5} {4,6} {2,7} {3,8}

{3,5} {5,6} {3,7} {5,8}

{4,5} {4,5,6} {4,7} {6,8}

{3,4,5} {5,7} {7,8}

{6,7} {3,7,8}

{3,5,7} {5,6,8}

{4,5,7} {5,7,8}

{4,6,7} {6,7,8}

{5,6,7} {5,6,7,8}

{4,5,6,7}

Note that these subsets do not all generate numerical semigroups, as their GCD is unrestricted, cf. A358392. The complement is counted by A365046, first differences of A364914.

(End)

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.

We define the (run-) compression of a sequence to be the anti-run obtained by reducing each run of repeated parts to a single part. Alternatively, compression removes all parts equal to the part immediately to their left. For example, (1,1,2,2,1) has compression (1,2,1).

The image first differs from A320340, A364347, A350838 in containing a(150) = 65.

The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.

The section-sum partition (A381436) of a multiset or partition y is defined as follows: (1) determine and remember the sum of all distinct parts, (2) remove one instance of each distinct part, (3) repeat until no parts are left. The remembered values comprise the section-sum partition. For example, starting with (3,2,2,1,1) we get (6,3).

Equivalently, the k-th part of the section-sum partition is the sum of all (distinct) parts that appear at least k times. Compare to the definition of the conjugate of a partition, where we count parts >= k.

The conjugate of a section-sum partition is a Look-and-Say partition; see A048767, union A351294, count A239455.

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.

We define the run-compression of a sequence to be the anti-run obtained by reducing each run of repeated parts to a single part. Alternatively, run-compression removes all parts equal to the part immediately to their left. For example, (1,1,2,2,1) has run-compression (1,2,1).

Number of different values of A007947(m) when A056239(m) is equal to n.

From Gus Wiseman, Sep 11 2023: (Start)

Also the number of finite sets of positive integers that can be linearly combined using all positive coefficients to obtain n. For example, the a(1) = 1 through a(7) = 12 sets are:

{1} {1} {1} {1} {1} {1} {1}

{2} {3} {2} {5} {2} {7}

{1,2} {4} {1,2} {3} {1,2}

{1,2} {1,3} {6} {1,3}

{1,3} {1,4} {1,2} {1,4}

{2,3} {1,3} {1,5}

{1,4} {1,6}

{1,5} {2,3}

{2,4} {2,5}

{1,2,3} {3,4}

{1,2,3}

{1,2,4}

Cf. A365073, A365311, A365312, A365322, A365380.

(End)

Also the number of integer partitions of n with distinct section-sums, where the k-th part of the section-sum partition is the sum of all (distinct) parts that appear at least k times. - Gus Wiseman, Apr 21 2025

The Look-and-Say partition of a multiset or partition y is obtained by interchanging parts with multiplicities. Hence, the multiplicity of k in the Look-and-Say partition of y is the sum of all parts that appear exactly k times. For example, starting with (3,2,2,1,1) we get (2,2,2,1,1,1), the multiset union of ((1,1,1),(2,2),(2)).

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798, sum A056239.

Also the number of ways to choose a set of disjoint strict integer partitions, one of each nonzero multiplicity in the prime factorization of n.

Includes all non-strict partitions (A047967).