1,1

The sequence exhibits phases involving alternating composite prime powers and squarefree semiprimes. These manifest in log-log scatterplot in a caustic fashion, where the composite prime power is very much larger than the squarefree semiprime for sufficiently large n.

Let P = the set of distinct prime divisors of j = a(n-1), and let Q = the set of distinct prime divisors of k = a(n). Let g = gcd(j, k) > 1 and let G = {P intersect Q}. Noncoprime j and k implies |G| > 0. This sequence is such that |G| > 0, |P| > |G|, and |Q| == |G|, or vice versa, yet neither j | k nor k | j.

Theorem: primes are prohibited. Proof: since we have gcd(j, k) > 1 and do not allow divisibility, and since primes must either divide or be coprime to another number m, primes do not appear in this sequence.

Theorem: squarefree semiprimes j = pq are followed by k = p^2 or k = q^2. Proof: since omega(j) = |P| = 2 and is squarefree, we have 2 cases pertaining to successor k, both with gcd(j, k) > 1.

1.) |P| == |G| implies |Q| > |G| and |Q| > |P|.

2.) |Q| == |G| implies |P| > |G| and |P| > |Q|.

The first case implies some prime r | k yet gcd(j, r) = 1. But this would require j | k, which is prohibited. The second case suggests either p | k or q | k, but so as to satisfy non-divisibility axiom, we are forced into either k = p^e, e > 1, or k = q^m, m > 1.

Corollary: powers of the same prime appear in natural order in this sequence.

There is a weaker alternation between numbers in A120944 and A350352 as n is sufficiently large. This alternation exhibits prime power factor features akin to the composite prime power-squarefree semiprime alternation.

Conjecture: permutation of composite numbers.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Michael De Vlieger, Annotated log log scatterplot of a(n), n = 1..1192, showing records in red and local minima in blue.

Michael De Vlieger, Log-log scatterplot of a(n), n = 1..2^16, showing records in red, local minima in blue, accentuating composite prime powers in green and squarefree semiprimes in gold.

Michael De Vlieger, Prime power factor study of a(n), n = 1..10^4 where n increases from left to right, and pi(p) from bottom to top in the top image. Multiplicity e of p^e is represented by black for e = 1, red for e = 2, and a color function to the maximum multiplicity in the range thereafter. On the lower bar we show composite prime powers in yellow, squarefree semiprimes in orange, numbers in A350352 in green, and those in A126706 in blue.

nn = 2^7; c[_] = 0; j = a[1] = 4; c[4] = 1; u = 6; Do[Set[k, u]; Set[m, PrimeNu[j]]; While[Nand[c[k] == 0, ! Divisible[#2, #1] & @@ Sort[{j, k}], And[#2 > #3, #1 == #3] & @@ Append[Sort[{m, PrimeNu[k]}], PrimeNu[GCD[j, k]]]], k++]; Set[{a[i], c[k]}, {k, i}]; j = k; If[k == u, While[Nand[c[u] == 0, CompositeQ@ u], u++]], {i, 2, nn}]; Array[a, nn]

Cf. A006881, A064413, A126706, A246547, A337687, A353916.

Sequence in context: A288814 A062711 A280739 * A117347 A088011 A289424

Adjacent sequences: A353914 A353915 A353916 * A353918 A353919 A353920

Michael De Vlieger, May 10 2022

approved