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A256393 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). 8
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
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
a(2*n) = A070229(a(2*n-1)); a(2*n+1) = A061228(a(2*n)). - Reinhard Zumkeller, May 06 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
LINKS
FORMULA
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.
MAPLE
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:
seq(a[i], i=1..100); # Robert Israel, May 03 2015
MATHEMATICA
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 *)
PROG
(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
(Scheme, with memoization-macro definec) (definec (A256393 n) (cond ((= 1 n) 2) ((even? n) (+ (A256393 (- n 1)) (A006530 (A256393 (- n 1))))) (else (+ (A256393 (- n 1)) (A020639 (A256393 (- n 1))))))) ;; Antti Karttunen, Apr 18 2015
(Haskell)
a256393 n = a256393_list !! (n-1)
a256393_list = 2 : zipWith ($) (cycle [a070229, a061228]) a256393_list
-- Reinhard Zumkeller, May 06 2015
CROSSREFS
Cf. A006530 (greatest prime factor), A020639 (least prime factor), A076271.
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).
Sequence in context: A143145 A328212 A352191 * A130664 A014011 A064424
KEYWORD
nonn
AUTHOR
Jan Guichelaar, Mar 28 2015
EXTENSIONS
More terms from Michel Marcus, Apr 02 2015
Replaced the name with more succinct description, moved old name to comments - Antti Karttunen, Apr 18-19 2015
STATUS
approved

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Last modified March 29 09:14 EDT 2024. Contains 371268 sequences. (Running on oeis4.)