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This sequence groups the reduced positive proper fractions p/q in a manner analogous to the Farey sequence (A005728, which adds the endpoints 0/1 and 1/1). However, instead of limiting the size of the denominator (q <= n as is done in the Farey sequence), we limit phi(q) = A000010(q), where phi is the Euler totient function. The computation requires A014197(r) = the number of natural numbers q such that phi(q) = r.

When k > 0, a(2k+1) = a(2k), because there are no natural numbers whose Euler totient equals 2k+1.

The sequences A_x were defined in A384669; please see that sequence for more details.

Because of the continuity of the functions g_k(x) defined in A384669, if k is a term of A_y, then k is a term of A_x with x rational sufficiently close to y; so it suffices to study A_x for x rational, 0 < x < 1.

Let x = p/q, with p and q natural numbers, p < q. Then B_n is the ordered union of the ranges of A_x, where q has Euler totient <= n (that is, A000010(q) <= n). B_1 is just the sequence A_(1/2), that is, A384669. The present sequence B_2 is the ordered union of the ranges of A_x, where x = 1/6, 1/4, 1/3, 1/2, 2/3, 3/4, 5/6.

8 is the first term in B_2 (this sequence) that is not in B_1 = A384669.

15120 appears to be the first term in B_4 that is not in B_2 (this sequence).

More discussion about B_n and B_oo appears in a linked pdf.

This sequence is the special case x = 1/2 of a class of sequences A_x indexed by real numbers x, defined as follows. Let k be a natural number with prime factorization k = Product_{i=1..r} (p_i)^(e_i), where p_i are distinct primes, and e_i are natural numbers. Define f_x(k) = Sum_{i=1..r} (e_i)^x. Then the sequence A_x consists of natural numbers k where f_x(k) sets a new record.

We may define g_k(x) = f_x(k), so that g_k consists of real functions indexed by natural numbers. Each g_k is a sum of exponential functions, whose bases are natural numbers.

We provide facts, conjectures, and questions about the class of sequences A_x.

Facts:

For all real x, A_x is an infinite, increasing sequence of numbers of least prime signature (A025487) starting with 1 and 2.

For all x <= 0, A_x coincides with the primorials (A002110). For all x >= 1, A_x coincides with the powers of 2 (A000079). Thus the interesting sequences A_x have 0 <= x <= 1.

For all x > 0, A_x(n+1) <= 2*A_x(n).

1, 2, 4, 6, 12, 24, and no other natural numbers, are terms of every A_x with 0 < x < 1.

Conjectures:

The sequences A_x appear to be distinct in the following sense. Given 0 <= x < y <= 1, not only are A_x and A_y distinct, their ranges appear to have finite intersection.

Define A_0+, respectively A_1- as the limit of A_x as x approaches 0 from the right, respectively as x approaches 1 from the left. Then A_0+ appears to coincide with A168264, and A_1- appears to coincide with A029744 without the term 3.

If 0 < x < 1 and d is a natural number, then d divides some term of A_x.

If 0 < x < 1 and d is a natural number, then d divides infinitely many terms of A_x.

If 0 < x < 1 and d is a natural number, then d divides all sufficiently large terms of A_x.

Define S_n to be the set of x for which A_x(n) achieves a minimum value as a function of x (well-defined because the values are integers). Then for fixed n >= 10, S_n appears to be a subset of the interval log(2)/log(3) <= x <= log(3)/log(4), although in general S_n itself is not an interval.

Questions:

Does every number of least prime signature appear in at least one A_x for some x? 216 is not a term in any A_x I have examined.

Are there any x for which A_x contains infinitely many highly composite numbers (A002182), respectively infinitely many deeply composite numbers (A095848)?

Does the sequence of sets S_n have any limit points x in [0,1]? If x is such a limit point, A_x would presumably grow more slowly than other A_x. Examining A_x for x rational with denominator 720, A_x appears to contain a maximum number of terms (80) less than 2 * 10! when x = 507/720, 511/720, 515/720, and 517/720, all between 0.70 and 0.72.

Answer to the first question above: No, 216 is not a term of A_x for any x. If it were, we would have g_216(x) > g_210(x) and g_216(x) > g_192(x), i.e., 2*3^x > 4 and 2*3^x > 6^x+1. The first inequality holds only if x > x0 = log(2)/log(3). But 2*3^x0 < 6^x0+1 and it is easily verified that the function 2*3^x - (6^x+1) is decreasing for x > x0, so the second equality cannot hold when x > x0. - Pontus von Brömssen, Jun 13 2025

Additional facts from Hal M. Switkay, Jun 29 2025: (Start)

If k1 and k2 have the same prime signature, then the functions g_k1(x) and g_k2(x) are identical. Hence g_k(x) is identical to g_A046523(k)(x), where A046523(k) is the smallest number with the same prime signature as k. This is why the terms of A_x are all numbers of least prime signature.

For all k, lim_{x->-oo} g_k(x) = A056169(k) = the number of unitary prime divisors of k.

For all k, g_k(0) = omega(k) = A001221(k) = the number of distinct prime divisors of k.

For all k, g_k(1) = bigomega(k) = A001222(k) = the number of prime divisors of k counted with multiplicity.

For all k > 1, lim_{x->oo} [log(g_k(x))/x] = log(A051903(k)), where A051903(k) = the maximum exponent in the prime factorization of k. When k has least prime signature, A051903(k) = A007814(k), the exponent of the largest power of 2 dividing k. (End)

Using the notation of A385722, this sequence is B_1. - Hal M. Switkay, Jul 27 2025

This sequence is an involutory (period 2) permutation of the natural numbers. Its graph is symmetric in the line a(n) = n. It fixes all numbers with self-conjugate prime signatures (A384084), including 1 and the primes (A000040). It exchanges prime squares (A001248) and products of two distinct primes (A006881). It exchanges powers of 2 (A000079) with primorials (A002110).

The implied partition corresponding to k is the partition of bigomega(k) (A001222) formed by the prime exponents. For example, bigomega(18) = 3, which is partitioned as 2 + 1, because 18 = (3^2)(2^1), and 2 + 1 is a self-conjugate partition of 3. In contrast, while bigomega(42) = 3, 3 is partitioned as 1 + 1 + 1, because 42 = (2^1)(3^1)(7^1), and 1 + 1 + 1 is not a self-conjugate partition of 3.

This sequence is very similar to, but ultimately different from, A212166. The first difference is a(342) = 1083, whereas A212166(342) = 1080.

This sequence is a subsequence of A212166.

It includes 1 (empty partition) and all primes (A000040: partition 1), as well as numbers of the form (p^2)q, where p and q are distinct primes (A054753: partition 2 + 1).

k is a term in this sequence if and only if A046523(k) is a term in A181825.

This sequence properly includes the deeply composite numbers (A095848). 36 and 180 are the first terms of this sequence that are not deeply composite.

All terms in this sequence are numbers of least prime signature (A025487). Therefore it is easier to search for new terms in the product of A061394 (omega of least prime signature) and A036041 (bigomega of least prime signature). Similar to but ultimately different from A378630. Terms appear to be products of primorials (A002110) with powers of 2 (A000079), and thus are never divisible by the square of an odd prime.

The graph of this sequence gives it the appearance of a ruler-like function. If n is odd, a(n) = 1. If n is even and not a multiple of 6, a(n) = 2. If n is a multiple of 6 but not of 12, a(n) = 6, and so on.

Conway and Sloane identify 2 conjugacy classes of maximal finite irreducible subgroups of GL_5(Z). The 2 maximal groups are: 1) the wreath fifth power of the group of order 2, the automorphism group of Z^5, D5 and its dual, of order 3840; 2) the product of the symmetric group of degree 6 with the group of order 2, the automorphism group of the A5 lattice and its dual, with order 1440.

Conway and Sloane identify 5 conjugacy classes of maximal finite irreducible subgroups of GL_4(Z). Of these, 2 are isomorphic to subgroups of other groups in the list. The 3 maximal groups are: 1) the Weyl group of F4, the automorphism group of the D4 lattice, with order 1152; 2) the wreath square of the dihedral group of order 12, the automorphism group of the (A2)^2 lattice, with order 288; 3) the product of the symmetric group of degree 5 with the group of order 2, the automorphism group of the A4 lattice (and its dual), with order 240.