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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

The fixed points of the x -> A181819(x) map are 1 and 2. Note that the x -> A000005(x) map has the same fixed points, and that A000005(n) = A181819(n) iff n is cubefree (cf. A004709). Under the x -> A181819(x) map, it seems significantly easier to generalize about which kinds of integers take a given number of iterations to reach a fixed point than under the x -> A000005(x) map.

Also the number of steps in the reduction of the multiset of prime factors of n wherein one repeatedly takes the multiset of multiplicities. For example, the a(90) = 5 steps are {2,3,3,5} -> {1,1,2} -> {1,2} -> {1,1} -> {2} -> {1}. - Gus Wiseman, May 13 2018

This sequence and A000028 (its complement) give the unique solution to the problem of splitting the positive integers into two classes in such a way that products of pairs of distinct elements from either class occur with the same multiplicities [Lambek and Moser]. Cf. A000069, A001969.

See A000028 for precise definition, Maple program, etc.

The sequence contains products of even number of distinct terms of A050376. - Vladimir Shevelev, May 04 2010

From Vladimir Shevelev, Oct 28 2013: (Start)

Numbers m such that the infinitary Möbius function (A064179) of m equals 1. (This follows from the definition of A064179.)

A number m is in the sequence iff the number k = k(m) of terms of A050376 that divide m with odd maximal exponent is even (see example).

(End)

Numbers k for which A064547(k) [or equally, A268386(k)] is even. Numbers k for which A010060(A268387(k)) = 0. - Antti Karttunen, Feb 09 2016

The sequence is closed under the commutative binary operation A059897(.,.). As integers are self-inverse under A059897(.,.), it therefore forms a subgroup of the positive integers considered as a group under A059897(.,.). Specifically (expanding on the comment above dated May 04 2010) it is the subgroup of even length words in A050376, which is the group's lexicographically earliest ordered minimal set of generators. A000028, the set of odd length words in A050376, is its complementary coset. - Peter Munn, Nov 01 2019

From Amiram Eldar, Oct 02 2024: (Start)

Numbers whose number of infinitary divisors (A037445) is a square.

Numbers whose exponentially odious part (A367514) has an even number of distinct prime factors, i.e., numbers k such that A092248(A367514(k)) = 0. (End)

A prime metasignature is analogous to the signature of a partition (cf. A115621); it is the signature of a prime signature.

Row n also gives prime signature of A181819(n).

Old name: 'Positions of zeros in A268387: numbers n such that when the exponents e_1 .. e_k in their prime factorization n = p_1^e_1 * ... * p_k^e_k are bitwise-xored together, the result is zero.

From Peter Munn, Sep 14 2019 and Dec 01 2019: (Start)

When trailing zeros are removed from the terms written in base p, for any prime p, every positive integer not divisible by p appears exactly once. This is the lexicographically earliest sequence with this property.

The closure of A238748 with respect to the commutative binary operation A059897(.,.). As integers are self-inverse under A059897(.,.), the sequence thereby forms a subgroup, denoted H, of the positive integers under A059897(.,.). H is a subgroup of A000379.

(The symbol ^ can take on a meaning in relation to a group operation. However, in this comment ^ denotes the power operator for standard integer multiplication.) For any prime p, the subgroup {p^k : k >= 0} and H are each a (left and right) transversal of the other. For k >= 0 and primes p_1 and p_2, the cosets (p_1^k)H and (p_2^k)H are the same.

(End)

From Peter Munn, Dec 01 2021: (Start)

If we take the square root of the square terms we reproduce the sequence itself. The set of all products of a square term and a squarefree term is the sequence as a set.

The terms are the elements of the ideal generated by {6} in the ring defined in A329329. Similarly, the ideal generated by {8} gives A262675. 6 and 8 are images of each other under A225546(.), which is an automorphism of the ring. So this sequence and A262675, as sets, are images of each other under A225546(.). The elements of the ideal generated by {6,8} form the notable set A000379.

(End)

The numbers of terms not exceeding 10^k, for k = 1, 2, ..., are 3, 35, 326, 3275, 33090, 332435, 3327555, 33283964, 332868092, 3328794682, ... . Apparently, the asymptotic density of this sequence exists and equals 0.3328... . - Amiram Eldar, Nov 28 2023

Numbers n such that 2^k - (2*n+1) and (2*n+1)*2^k - 1 are both prime:

For k = 0: 2, 3, 5, 7, 13, 17, ... Intersection of A000043 and A000043

for k = 1: 3, 4, 6, 94, ... Intersection of A050414 and A002235

for k = 2: 4, 8, 10, 12, 18, 32, ... Intersection of A059608 and A001770

for k = 3: Intersection of A059609 and A001771

for k = 4: 21, ... Intersection of A059610 and A002236

for k = 5: Intersection of A096817 and A001772

for k = 6: Intersection of A096818 and A001773

for k = 7: 5, 10, 14, ... Intersection of A059612 and A002237

for k = 8: 6, 16, 20, 36, ... Intersection of A059611 and A001774

for k = 9: 5, 21, ... Intersection of A096819 and A001775

for k = 10: 7, 13, ... Intersection of A096820 and A002238

for k = 11: 6, 8, 12, ...

for k = 12: 9, ...

for k = 13: 5, 8, 10, ...

First differs from A268390 in lacking 210.

First differs from A238748 in lacking 210.

These are the Heinz numbers of the partitions counted by A367588.