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A256393
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Start from a(1) = 2, then alternately add either the largest (if n is even), or the smallest (if n is odd) prime factor of the preceding term a(n-1) to get a(n).
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8
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2, 4, 6, 9, 12, 15, 18, 21, 24, 27, 30, 35, 40, 45, 48, 51, 54, 57, 60, 65, 70, 77, 84, 91, 98, 105, 108, 111, 114, 133, 140, 147, 150, 155, 160, 165, 168, 175, 180, 185, 190, 209, 220, 231, 234, 247, 260, 273, 276, 299, 312, 325, 330, 341, 352, 363, 366, 427
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OFFSET
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1,1
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COMMENTS
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After the initial term, each even-indexed term equals the preceding term plus its largest prime factor, and each odd-indexed term equals the preceding term plus its smallest prime factor.
See also sequence A076271 where a(n+1) = a(n) + lpf(a(n)).
Each term shares exactly one prime factor with the immediately preceding term, and because the sequence is strictly increasing, all the terms after 2 are composite. - Antti Karttunen, Apr 19 2015
From a(3) onward, the terms are alternately even and odd. - Jan Guichelaar, Apr 24 2015
For prime p let [p] denote the sequence with a(1)=p, and generated as for the terms of the current sequence (which according to this notation is then the same as [2]. It so happens that the sequence [p] (for any p?) merges with [2] sooner or later, taking the form of a "tree" as shown in the attached image (Including prime starts up to p=67). Is this pattern of merging bounded or not? Is there just one tree or are there many? Interesting to speculate. The numbers corresponding to the arrival points in [2] of [p] is the sequence 2,6,15,21,51,57,77,84.... The sequence of ("excluded")numbers which do not arise in [p] for any prime p starts as 8,16,20,25,28,32,36,44... Other sequences may refer to the number of iterations required to merge [p] into [2]. See tree picture. - David James Sycamore, Aug 25 2016
In this picture, one could also include some [c] sequences, with composite c, see A276269. - Michel Marcus, Aug 26 2016
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LINKS
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FORMULA
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a(1) = 2; a(2n) = a(2n-1) + gpf(a(2n-1)), a(2n+1) = a(2n) + lpf(a(2n)), where gpf = greatest prime factor = A006530, lpf = least prime factor = A020639.
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MAPLE
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a[1]:= 2;
for n from 2 to 100 do
if n::even then a[n]:= a[n-1] + max(numtheory:-factorset(a[n-1]))
else a[n]:= a[n-1] + min(numtheory:-factorset(a[n-1]))
fi
od:
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MATHEMATICA
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f[n_] := Block[{pf = First /@ FactorInteger@ n}, If[EvenQ@ n, Max@ pf, Min@ pf]]; s = {2}; lmt = 58; For[k = 2, k <= lmt, k++, AppendTo[s, s[[k - 1]] + f@ s[[k - 1]]]]; s (* Michael De Vlieger, Apr 19 2015 *)
FoldList[Function[f, If[EvenQ@ #2, #1 + First@ f, #1 + Last@ f]][FactorInteger[#1][[All, 1]]] &, Range[2, 59]] (* Michael De Vlieger, Aug 26 2016 *)
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PROG
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(PARI) lista(nn) = {print1(a = 2, ", "); for (n=2, nn, f = factor(a); if (n % 2, a += f[1, 1], a += f[#f~, 1]); print1(a, ", "); ); } \\ Michel Marcus, Apr 02 2015
(Haskell)
a256393 n = a256393_list !! (n-1)
a256393_list = 2 : zipWith ($) (cycle [a070229, a061228]) a256393_list
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CROSSREFS
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Cf. A257244 (the first differences; the unique prime factors shared by each pair of successive terms), A257245, A257246 (their bisections), A257247 (numbers n such that GCD(a(2n-1),a(2n)) = GCD(a(2n),a(2n+1)), which is prime).
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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Replaced the name with more succinct description, moved old name to comments - Antti Karttunen, Apr 18-19 2015
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STATUS
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approved
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