

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(n1) 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
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OFFSET

1,1


COMMENTS

After the initial term, each evenindexed term equals the preceding term plus its largest prime factor, and each oddindexed 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*n1)); 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

Antti Karttunen, Table of n, a(n) for n = 1..4096
David Sycamore, Tree showing what happens if the initial 2 is replaced by a different prime (Corrected Aug 29 2016)


FORMULA

a(1) = 2; a(2n) = a(2n1) + gpf(a(2n1)), 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[n1] + max(numtheory:factorset(a[n1]))
else a[n]:= a[n1] + min(numtheory:factorset(a[n1]))
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 memoizationmacro 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 !! (n1)
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(2n1),a(2n)) = GCD(a(2n),a(2n+1)), which is prime).
Cf. A061228, A070229.
Sequence in context: A030763 A143145 A328212 * A130664 A014011 A064424
Adjacent sequences: A256390 A256391 A256392 * A256394 A256395 A256396


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 1819 2015


STATUS

approved



