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This is a permutation of all squarefree numbers > 1.

Let gpf(m) = A006530(m) and let phi(m) = A000010(m) for m in A005117.

Row n contains m in A005117 such that A000720(A006530(m)) = n, sorted such that phi(m)/m increases as k increases.

Let m be the squarefree kernel A007947(m') of m'. We only consider squarefree m since phi(m)/m = phi(m')/m'. Let prime p | n and prime q be a nondivisor of n.

Since m is squarefree, we might encode the multiplicities of its prime divisors in a positional notation M that is finite at n significant digits. For example, m = 42 can be encoded reverse(A067255(42)) = 1,0,1,1 = 7^1 * 5^0 * 3^1 * 2^1. It is necessary to reverse row m of A067255 (hereinafter simply A067255(m)) so as to preserve zeros in M = A067255(m) pertaining to small nondivisor primes q < p. The code M is a series of 0's and 1's since m is squarefree. Then it is clear that row n contains all m such that A067255(m) has n terms, and there are 2^(n - 1) possible terms for n >= 1.

We may use an approach that generates the binary expansion of the range 2^(n - 1) < M < 2^n - 1, or we may append 1 to the reversed (n - 1)-tuples of {1, 0} to achieve codes M -> m for each row n, which is tantamount to ordering according to A059894.

Originally it was thought that the codes M were in order of the latter algorithm, and we could avoid sorting. Observation shows that the m still require sorting by the function phi(m)/m indeed to be in increasing order in row n. Still, the latter approach is slightly more efficient than the former in generating the sequence.

Motivated by A362631, but instead of using one prime divisor of a(n-2) which does not divide a(n-1), the product of all such primes is used to compute a(n). - David James Sycamore, May 07 2023

From Michael De Vlieger, May 27 2023: (Start)

Primes p(k) enter the sequence in order and fairly regularly through a(20543) = p(15) = 47, immediately following primorial A002110(k-1). However, a(87723) = p(17) = 59 is the next prime to appear, following a(87722) = A002110(16).

Conjecture: all primes appear eventually, but not in order. (End)

Similar to A280866, except that the denominator here is rad(a(n-1)) instead of rad(a(n-2)). Also related to A369825. - David James Sycamore, Jan 27 2024

From Michael De Vlieger, Apr 23 2024: (Start)

Conjecture: permutation of natural numbers.

Conjecture: the smallest missing number is always either prime or a powerful number.

Primes do not appear in order; a(87723) = 59 but a(91307) = 53.

Powerful numbers appear in clusters, e.g., for n roughly between 91200 and 91320.

Though it appears primorials are always followed by primes, it is logically possible but rare that primorials can be followed by a composite number. (End)

Let gpf(m) = A006530(m) and let phi(m) = A000010(m) for m in A005117.

Row n contains m in A005117 such that A006530(m) = n, sorted such that phi(m)/m increases as k increases.

Let m be the squarefree kernel A007947(m') of m'. We only consider squarefree m since phi(m)/m = phi(m')/m'. Let prime p | n and prime q be a nondivisor of n.

Since m is squarefree, we might encode the multiplicities of its prime divisors in a positional notation M that is finite at n significant digits. For example, m = 42 can be encoded reverse(A067255(42)) = 1,0,1,1 = 7^1 * 5^0 * 3^1 * 2^1. It is necessary to reverse row m of A067255 (hereinafter simply A067255(m)) so as to preserve zeros in M = A067255(m) pertaining to small nondivisor primes q < p. The code M is a series of 0's and 1's since m is squarefree. Then it is clear that row n contains all m such that A067255(m) has n terms, and there are 2^(n - 1) possible terms for n >= 1.

We may use an approach that generates the binary expansion of the range 2^(n - 1) < M < 2^n - 1, or we may append 1 to the reversed (n - 1)-tuples of {1, 0} (as A059894) to achieve codes M -> m for each row n.

Originally it was thought that the codes M were in order of the latter algorithm, and we could avoid sorting. Observation shows that the m still require sorting by the function phi(m)/m indeed to be in increasing order in row n. Still, the latter approach is slightly more efficient than the former in generating the sequence.

This sequence interprets the code M as a binary value. The sequence is a permutation of the natural numbers since the ratio phi(m)/m is unique for squarefree m.

This sequence and A059894 are identical for 1 <= n <= 23.

Numbers of terms in rows n of this sequence and A059894 (partitioned by powers of 2) that are coincident: 1, 2, 4, 8, 14, 14, 10, 26, 14, 20, 10, 16, 22, 12, 18, 18, 16, 14, 18, 18, 18, 14, 16, ...}.

The graphs of this sequence and A059894 are similar.

The graph of this sequence feature squares of size 2^(j-1) at (x,y) = (h,h) where h = 2^j, integers, that have pi-radian rotational symmetry.