cp's OEIS Frontend

a(n) depends only on prime signature of n (cf. A025487). a(m) = a(n) iff m and n have the same prime signature, i.e., iff A046523(m) = A046523(n).

Because A046523 (the smallest representative of prime signature of n) and this sequence are functions of each other as A046523(n) = A181821(a(n)) and a(n) = a(A046523(n)), it implies that for all i, j: a(i) = a(j) <=> A046523(i) = A046523(j) <=> A101296(i) = A101296(j), i.e., that equivalence-class-wise this is equal to A101296, and furthermore, applying any function f on this sequence gives us a sequence b(n) = f(a(n)) whose equivalence class partitioning is equal to or coarser than that of A101296, i.e., b is then a sequence that depends only on the prime signature of n (the multiset of exponents of its prime factors), although not necessarily in a very intuitive way. - Antti Karttunen, Apr 28 2022

First term greater than 2 is a(360) = 3.

From Michel Marcus, Apr 24 2016: (Start)

A006939(n) gives the least m such that a(m) = n.

A062770 is the sequence of integers m such that a(m) = 1. (End)

We define the k-th omega of n to be Omega(red^{k-1}(n)) where Omega = A001222 and red^{k} is the k-th functional iteration of A181819. The first two omegas are A001222 and A001221, while this sequence is the third, and A323022 is the fourth. The zeroth omega is not uniquely determined from prime signature, but one possible choice is A056239 (sum of prime indices). - Gus Wiseman, Jan 02 2019

Sanna (2020) proved that for each k>=1, the sequence of numbers n with A071625(n) = k has an asymptotic density A_k = (6/Pi^2) * Sum_{n>=1, n squarefree} rho_k(n)/psi(n), where psi is the Dedekind psi function (A001615), and rho_k(n) is defined by rho_1(n) = 1 if n = 1 and 0 otherwise, rho_{k+1}(n) = 0 if n = 1 and (1/(n-1)) * Sum_{d|n, dAmiram Eldar, Oct 18 2020

First differs from A130091 (Wilf partitions) in having 216.

See A239455 for the definition of Look-and-Say partitions.

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.

First differs from A130092 (non-Wilf partitions) in lacking 216.

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 sequence of runs of a sequence consists of its maximal consecutive constant subsequences when read left-to-right. For example, the runs of (2,2,1,1,1,3,2,2) are (2,2), (1,1,1), (3), (2,2), with sums (4,3,3,4).

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.

This sequence represents the transformation f(P) described by Kimberling at A237685.

We define the omega-sequence of n to have length A323014(n), and the k-th term is Omega(red^{k-1}(n)), where Omega = A001222 and red^{k} is the k-th functional iteration of A181819.

Except for n = 1, all rows end with 1. If n is not prime, the term in row n prior to the last is A304465(n).

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 sequence of runs of a sequence consists of its maximal consecutive constant subsequences when read left-to-right. For example, the runs of (2,2,1,1,1,3,2,2) are (2,2), (1,1,1), (3), (2,2), with sums (4,3,3,4).

The adjusted frequency depth of an integer partition is 0 if the partition is empty, and otherwise it is one plus the number of times one must take the multiset of multiplicities to reach a singleton. For example, the partition (32211) has adjusted frequency depth 5 because we have: (32211) -> (221) -> (21) -> (11) -> (2).

The term "frequency depth" appears to have been coined by Clark Kimberling in A225485 and A225486, and can be applied to both integers (A323014) and integer partitions (this sequence).

Let S = {x(1),...,x(k)} be a multiset whose distinct elements are y(1),...,y(h). Let f(i) be the frequency of y(i) in S. Define F(S) = {f(1),..,f(h)}, F(1,S) = F(S), and F(m,S) = F(F(m-1),S) for m>1. Then lim(F(m,S)) = {1} for every S, so that there is a least positive integer i for which F(i,S) = {1}, which we call the frequency depth of S.

Equivalently, the frequency depth of an integer partition is the number of times one must take the multiset of multiplicities to reach (1). For example, the partition (32211) has frequency depth 5 because we have: (32211) -> (221) -> (21) -> (11) -> (2) -> (1). - Gus Wiseman, Apr 19 2019

From Clark Kimberling, Sep 26 2023: (Start)

Below, m^n abbreviates the sum m+...+m of n terms. In the following list, the numbers p_1,...,p_k are distinct, m >= 1, and k >= 1. The forms of the partitions being counted are as follows:

column 1: [n],

column 2: [m^k],

column 3: [p_1^m,...,p_k^m],

column 4: [(p_1^m_1)^m,..., (p_k^m_k)^m], distinct numbers m_i.

Column 3 is of special interest. Assume first that m = 1, so that the form of partition being counted is p = [p_1,...,p_k], with conjugate given by [q_1,...,q_m] where q_i is the number of parts of p that are >= i. Since the p_i are distinct, the distinct parts of q are the integers 1,2,...,k. For the general case that m >= 1, the distinct parts of q are the integers m,...,km. Let S(n) denote the set of partitions of n counted by column 3. Then if a and b are in the set S*(n) of conjugates of partitions in S(n), and if a > b, then a - b is also in S*(n). Call this the subtraction property. Conversely, if a partition q has the subtraction property, then q must consist of a set of numbers m,..,km for some m. Thus, column 3 counts the partitions of n that have the subtraction property. (End)

The function A181819 maps p^i*...*q^j to prime(i)*...*prime(j) where p through q are distinct primes.