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A288313
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Let b(k) denote A056240(k); the sequence lists numbers b(2*n) where for all m > n, b(2*m) > b(2*n).
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9
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2, 4, 8, 15, 21, 33, 39, 51, 57, 69, 87, 93, 111, 123, 129, 141, 159, 177, 183, 201, 213, 219, 237, 249, 267, 291, 303, 309, 321, 327, 339, 381, 393, 411, 417, 447, 453, 471, 489, 501, 519, 537, 543, 573, 579, 591, 597, 633, 669, 681, 687, 699, 717, 723, 753, 771, 789, 807, 813, 831, 843
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OFFSET
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1,1
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COMMENTS
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This is an ascending subsequence of A056240 with even argument terms.
After the first three (even) terms, a(1) = b(2) = 2, a(2) = b(4) = 4, a(3) = b(6) = 8 respectively, all subsequent terms are odd (semiprime) numbers of the form 3*r, for r = primes 5, 7, 11, 13, .... The graph of all odd-valued terms a(n) for n >= 4 is a straight line (y = 3*x - 9), corresponding to b(2*n) = 3*(2*n) - 9 = 3*(2*n - 3) = 3*r, where r = 2*n - 3 is prime, and n is in sequence A098090. The sequence a(n) for n >= 4 is identical term for term to A001748(n) for n >= 3. In other words, for n >= 4, a(n) = 3*A000040(n-1).
If, for any even number n >= 6, n - 3 is prime, then A056240(n) belongs to this sequence.
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LINKS
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FORMULA
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a(1) = 2, a(2) = 4, a(3) = 8, and for n >= 4, a(n) = 3*A000040(n-1).
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EXAMPLE
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a(1) = 2 is included because for all n > 1, b(2n) > 2; likewise a(2) = b(4) = 4, and a(3) = b(6) = 8 are included. The first odd term, a(4) = b(8) = 15, is included since for all n > 4, b(2n) > 15. b(12) = 35 is not in this sequence because b(14) = 33 < 35, and only ascending terms are permitted.
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MATHEMATICA
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Join[{2, 4, 8}, 3*Prime[Range[3, 100]]] (* Paolo Xausa, Apr 16 2024 *)
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CROSSREFS
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KEYWORD
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nonn,changed
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AUTHOR
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EXTENSIONS
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Offset changed to 1 and entry edited to reflect this change by Michel Marcus, Jul 03 2017
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STATUS
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approved
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