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A366047
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Starting with a(1) = 2, the lexicographically earliest infinite sequence of distinct positive integers such that |a(n) - a(n-1)| is a divisor of a(n)*a(n-1), where |a(n) - a(n-1)| is not a prime and greater than 1.
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1
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2, 6, 10, 14, 18, 9, 36, 12, 4, 8, 16, 20, 24, 15, 30, 21, 42, 28, 32, 40, 44, 22, 26, 52, 48, 39, 78, 60, 35, 70, 45, 54, 27, 108, 72, 56, 64, 68, 34, 38, 76, 80, 55, 110, 66, 33, 132, 84, 63, 90, 65, 130, 104, 88, 92, 46, 50, 25, 150, 75, 100, 96, 87, 174, 58, 62, 124, 116, 112, 98, 49, 392
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OFFSET
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1,1
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COMMENTS
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For the sequence to be infinite no term can be a prime except for a(1) = 2. One can show that if a(n) is a prime p, then the only possible value for a(n-1) is 2p or p + p^2 since, if a term is prime, the preceding term must be a multiple of that prime. However the preceding term cannot be 2p since the difference between the terms would then be prime, therefore it must be p + p^2. However the only possible value for the term after a prime p is likewise p + p^2, but that has already been used, thus allowing a term to be prime would terminate the sequence.
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LINKS
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Michael De Vlieger, Log log scatterplot of a(n) n = 1..1024, showing prime powers in gold, squarefree numbers in green, and numbers neither squarefree nor prime powers in blue, highlighting powerful numbers that are not prime powers in light blue.
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EXAMPLE
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a(9) = 4 as |4 - a(8)| = |4 - 12| = 8, and 8 is a divisor of 4*12 = 48 and is not a prime. Note that |3 - 12| = 9 is a divisor of 3*12 = 36 and is not a prime, but as shown above a prime term will terminate the sequence so is not permitted.
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MATHEMATICA
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nn = 120; c[_] := False; s = {2, 6};
f[x_] := Times @@ FactorInteger[x][[All, 1]];
MapIndexed[Set[{a[First[#2]], c[#1]}, {#1, True}] &, s];
Set[{j, u}, {s[[-1]], 4}];
Do[k = u;
While[Or[c[k], #1 < 4, PrimeQ[#1],
! Divisible[j*k, #1], ! Divisible[j, #2], ! Divisible[k, #2]] & @@
{#, f[#]} &@ Abs[j - k], k++];
Set[{a[n], c[k], j}, {k, True, k}];
If[k == u, While[Or[c[u], PrimeQ[u]], u++]], {n, Length[s] + 1, 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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