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A354720
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a(n) = n for n <= 3; let i = a(n-2) and j = a(n-1); a(n+1) = least k not already in the sequence such that (j, k) = 1 and (i, k) = m > 1 and only one of either omega(i) or omega(k) exceed omega(m), where omega = A001221.
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1
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1, 2, 3, 10, 21, 4, 7, 6, 35, 8, 5, 12, 55, 9, 11, 15, 22, 25, 16, 45, 14, 27, 32, 33, 20, 81, 64, 39, 28, 13, 42, 65, 18, 125, 66, 85, 24, 17, 30, 119, 36, 49, 60, 77, 40, 121, 70, 99, 50, 231, 128, 63, 26, 105, 169, 75, 52, 165, 182, 135, 56, 195, 154, 117, 44
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OFFSET
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1,2
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COMMENTS
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Theorem: even terms cannot be adjacent. Proof: If prime p | j, then p cannot divide k as well, because then (j, k) >= p and by definition of "prime", p > 1, which contradicts the axiom (j, k) = 1. Since 2 is prime, consecutive even terms are prohibited.
A restriction on the Yellowstone sequence A098550 analogous to A353916 regarding its relationship to A064413.
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LINKS
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Michael De Vlieger, Annotated log-log scatterplot of a(n), n = 1..2^14, showing records in red, local minima in blue, highlighting fixed points in gold, primes in green, and composite prime powers in light green.
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MATHEMATICA
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nn = 120; s = Range[3]; state = {2, 3, 4, 7}; u = 1; c[_] = 0; f[j_, k_] := Which[j == k, 5, GCD[j, k] == 1, 0, True, 1 + FromDigits[Map[Which[Mod[##] == 0, 1, PowerMod[#1, #2, #2] == 0, 2, True, 0] & @@ # &, Permutations[{k, j}]], 3]]; Array[Set[{a[s[[#]]], c[#]}, {#, s[[#]]}] &, Length[s]]; While[c[u] > 0, u++]; Set[{i, j}, s[[-2 ;; -1]]]; Do[k = u; While[Nand[c[k] == 0, MemberQ[state, f[i, k]], CoprimeQ[j, k]], k++]; Set[{a[n], c[k], i, j}, {k, n, j, k}]; If[k == u, While[c[u] > 0, u++]], {n, Length[s] + 1, 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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