cp's OEIS Frontend

First differs from A050326 at a(30) = 4, A050326(30) = 5.

First differs from A339742 at a(42) = 5, A339742(42) = 4.

First differs from A381441 at a(30) = 4, A381441(30) = 5.

First differs from A381633 at a(210) = 10, A381633(210) = 12.

Also the number of multisets that can be obtained by taking the sums of prime indices of each factor in a factorization of n into squarefree numbers > 1 with distinct sums of prime indices (A056239).

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.

A multiset partition con be regarded as an arrow in the ranked poset of integer partitions. For example, we have {{1},{1,2},{1,3},{1,2,3}}: {1,1,1,1,2,2,3,3} -> {1,3,4,6}, or (33221111) -> (6431) (depending on notation).

Set multipartitions with distinct block-sums are generally not transitive. For example, we have arrows: {{1},{1,2}}: {1,1,2} -> {1,3} and {{1,3}}: {1,3} -> {4}, but there is no arrow {1,1,2} -> {4}.

Length of row n equals A001221(n).

The multiset of positive exponents in n's prime factorization completely determines a(n) for a host of OEIS sequences, including several "core" sequences. Of those not cross-referenced here or in A212172, many can be found by searching the database for A025487.

(Note: Differing opinions may exist about whether the prime signature of n should be defined as this multiset itself, or as a symbol or collection of symbols that identify or "signify" this multiset. The definition of this sequence is designed to be compatible with either view, as are the original comments. When n >= 2, the customary ways to signify the multiset of exponents in n's prime factorization are to list the constituent exponents in either nonincreasing or nondecreasing order; this table gives the nonincreasing version.)

Table lists exponents in the order in which they appear in the prime factorization of a member of A025487. This ordering is common in database comments (e.g., A008966).

Each possible multiset of an integer's positive prime factorization exponents corresponds to a unique partition that contains the same elements (cf. A000041). This includes the multiset of 1's positive exponents, { } (the empty multiset), which corresponds to the partition of 0.

Differs from A124010 from a(23) on, corresponding to the factorization of 18 = 2^1*3^2 which is here listed as row 18 = [2, 1], but as [1, 2] (in the order of the prime factors) in A124010 and also in A118914 which lists the prime signatures in nondecreasing order (so that row 12 = 2^2*3^1 is also [1, 2]). - M. F. Hasler, Apr 08 2022

These are points at which the second differences (A376590) are zero.

If a(29) is not 0, then it is > 10^12. - Lucas A. Brown, Oct 25 2024

First differs from A240309 at a(11) = 23, A240309(11) = 25.

First differs from A381990 at a(17) = 126, A381990(17) = 127.

We call a multiset normal iff it covers an initial interval of positive integers. The size of a multiset is the number of elements, counting multiplicity.

An integer partition is non-multigraphical if it does not comprise the multiset of vertex-degrees of any multigraph (multiset of non-loop edges). Multigraphical partitions are counted by A209816, non-multigraphical partitions by A000070.

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

The following are equivalent characteristics for any positive integer n:

(1) the multiset of prime indices of n can be partitioned into strict pairs (a multiset of edges);

(2) n can be factored into squarefree semiprimes;

(3) the unordered prime signature of n is multigraphical.

Row-lengths are A051903.

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.

The section-sum partition 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.

First differs from A212167 in having 3600.

First differs from A335433 in lacking 72.

First differs from A339741 in having 1080.

First differs from A345172 in lacking 72.

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.

Also numbers that can be written as a product of squarefree numbers with distinct sums of prime indices.