A352097 - OEIS

%I #14 Jun 09 2022 10:02:57

%S 4,9,10,21,8,15,14,25,6,35,16,45,22,27,20,33,26,51,28,39,32,57,34,63,

%T 38,49,12,77,18,55,24,65,36,85,42,95,44,75,46,81,40,69,50,87,52,93,56,

%U 99,58,105,62,91,30,119,48,133,54,161,60,203,64,145,66,115,68

%N 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.

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

%C 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.

%C All i and k must have an "unrelated" (neither coprime nor divisor, i.e., neutral) relationship in the sense of A045763.

%C 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}.

%C 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.

%C Composite quasi-rays in the Yellowstone sequence scatterplot are retained, bifurcated according to parity for same reasons as in that sequence.

%C Conjecture: permutation of the composite numbers.

%H Michael De Vlieger, <a href="/A352097/b352097.txt">Table of n, a(n) for n = 1..10000</a>

%H Michael De Vlieger, <a href="/A352097/a352097.png">Annotated log-log scatterplot of a(n)</a>, n = 1..2^10, showing records in red and local minima in blue.

%H Michael De Vlieger, <a href="/A352097/a352097_1.png">Prime power factor diagram for p^e | a(n)</a>, 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.

%t 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]

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

%K nonn

%O 1,1

%A _Michael De Vlieger_, Jun 03 2022