OFFSET
1,1
COMMENTS
Theorem: the sequence is in composites. Proof: Prime k must either divide or be coprime to i, but 1 < gcd(i, k) != k precludes k | i. Therefore there is no avenue for primes in the sequence.
All i and k must have an "unrelated" (neither coprime nor divisor, i.e., neutral) relationship in the sense of A045763.
Theorem: if prime p | j then p does not divide k. Consequence of coprimality axiom gcd(j, k) = 1. Hence, even terms are nonadjacent in the sequence. Therefore we begin this sequence with {4, 9}.
A version of Yellowstone sequence S = A098550 that strips out features attributable to prime S(n) and their cototient successors S(n+2). In S, those 2 groups produce quasi-rays in scatterplot that have primes appear late and their successors early in S.
Composite quasi-rays in the Yellowstone sequence scatterplot are retained, bifurcated according to parity for same reasons as in that sequence.
Conjecture: permutation of the composite numbers.
LINKS
Michael De Vlieger, Table of n, a(n) for n = 1..10000
Michael De Vlieger, Annotated log-log scatterplot of a(n), n = 1..2^10, showing records in red and local minima in blue.
Michael De Vlieger, Prime power factor diagram for p^e | a(n), n = 1..320, where the upper portion plots p^e at (n, pi(p)), with a color function representing e as follows: black = 1, red = 2, etc. to magenta representing the largest e in the range. The lower portion classifies squarefree semiprimes in orange, composite prime powers in yellow, otherwise squarefree numbers in green, and all other composites in blue.
MATHEMATICA
c[_] = 0; MapIndexed[Set[{a[First[#2]], c[#1]}, {#1, First[#2]}] &, {4, 9}]; Set[{i, j, u, nn}, {a[1], a[2], 6, 120}]; Do[k = u; While[Nand[c[k] == 0, And[# > 1, # != i, # != k] &@ GCD[i, k], CoprimeQ[j, k]], k++]; Set[{a[n], c[k], i, j}, {k, n, j, k}]; If[k == u, While[Nand[c[u] == 0, CompositeQ[u]], u++]], {n, 3, nn}]; Array[a, nn]
CROSSREFS
KEYWORD
nonn
AUTHOR
Michael De Vlieger, Jun 03 2022
STATUS
approved