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A352097
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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.
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1
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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
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OFFSET
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1,1
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COMMENTS
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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.
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LINKS
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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.
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MATHEMATICA
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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]
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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