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A357992
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a(1)=1,a(2)=2,a(3)=3. Thereafter, if there are prime divisors p of a(n-2) which do not divide a(n-1), a(n) is the least novel multiple of any such p. Otherwise a(n) is the least novel multiple of the squarefree kernel of a(n-2).
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1
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1, 2, 3, 4, 6, 8, 9, 10, 12, 5, 14, 15, 7, 18, 21, 16, 24, 20, 27, 22, 30, 11, 25, 33, 35, 36, 28, 39, 26, 42, 13, 32, 52, 34, 65, 17, 40, 51, 38, 45, 19, 48, 57, 44, 54, 55, 46, 50, 23, 56, 69, 49, 60, 63, 58, 66, 29, 62, 87, 31, 72, 93, 64, 75, 68, 70, 85, 74
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OFFSET
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1,2
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COMMENTS
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In other words, if a(n-2) has k prime divisors p_j; 1 <= j <= k which do not divide a(n-1), where 1 <= k <= omega(a(n-2)), and if m_j*p_j is the least multiple of p_j which is not already a term, then a(n) = Min{m_j*p_j; 1 <= j <= k}. Otherwise, every prime divisor of a(n-2) also divides a(n-1), in which case a(n) is the least multiple of the squarefree kernel of a(n-2) which is not already a term.
Unlike in A064413, a prime p occurrence here is not directly flanked by multiples of p, but by numbers x, y sharing divisors other than p. The terms preceding and following x and y are divisible by p. Typically we observe m*p, x, p, y, r*p, where gcd(x, y) > 1, for multiples m,r of p which do not always follow the (2,3) pattern observed in A064413 and elsewhere. The case of p = 13 is remarkable for being preceded and followed by two multiples of itself (the first five multiples of 13 occur within the span of eight consecutive terms).
Conjecture 1: A permutation of the positive integers with primes in natural order.
Conjecture 2: The primes are the slowest numbers to appear (see also A352187).
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LINKS
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Michael De Vlieger, Log-log scatterplot of a(n), n = 1..2^12 labeling records in red, local minima in blue, highlighting primes in green and other prime powers in gold.
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EXAMPLE
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With a(2)=2, and a(3)=3, a(4) must be 4, the least unused multiple of 2.
Likewise, with a(3),a(4) = 3,4 a(5) must be the 6, the least unused multiple of 3.
Since every divisor of 4 also divides 6 a(6) = 8, the least unused multiple of 2, (squarefree kernel of 4).
Since a(8),a(9) = 10,12 and 5 is the only prime dividing 10 but not 12, it follows that a(10) = 5.
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MATHEMATICA
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nn = 68; c[_] = False; q[_] = 1; Array[Set[{a[#], c[#]}, {#, True}] &, 2]; q[2] = 2; Do[m = FactorInteger[a[n - 1]][[All, 1]]; f = Select[m, CoprimeQ[#, a[n - 2]] &]; If[Length[f] == 0, While[Set[k, # * q[#]]; c[k], q[#]++] &[Times @@ m], Set[{k, q[#1]}, {#2, #2/#1}] & @@ First@ MaximalBy[Map[{#, Set[g, q[#]]; While[c[g #], g++]; # g} &, f], Last] ]; Set[{a[n], c[k]}, {k, True}], {n, 3, nn}]; Array[a, nn] (* Michael De Vlieger, Oct 23 2022 *)
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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