A352097 a(1) = 4, a(2) = 9; let i = a(n-2) and j = a(n-1); a(n) = least k not already in the sequence such that gcd(j, k) = 1 and 1 < gcd(i, k) != i != k.

4, 9, 10, 21, 8, 15, 14, 25, 6, 35, 16, 45, 22, 27, 20, 33, 26, 51, 28, 39, 32, 57, 34, 63, 38, 49, 12, 77, 18, 55, 24, 65, 36, 85, 42, 95, 44, 75, 46, 81, 40, 69, 50, 87, 52, 93, 56, 99, 58, 105, 62, 91, 30, 119, 48, 133, 54, 161, 60, 203, 64, 145, 66, 115, 68

Offset: 1

Author

Michael De Vlieger, Jun 03 2022

Keywords

• nonn

Comments

Variant of A098550 analogous to A240024 and its relationship to A064413.

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.

Crossrefs

Cf. A002808, A045763, A064413, A098550, A240024.

Programs

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]