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A109890 a(1)=1; for n>1, a(n) is the smallest number not already present which is a divisor or a multiple of a(1)+...+a(n-1). 14
1, 2, 3, 6, 4, 8, 12, 9, 5, 10, 15, 25, 20, 24, 16, 32, 48, 30, 18, 36, 27, 13, 7, 53, 106, 265, 159, 318, 212, 14, 107, 321, 214, 428, 642, 535, 35, 21, 181, 11, 33, 22, 23, 59, 70, 28, 151, 29, 19, 233, 466, 2563, 699, 932, 40, 26, 38, 31, 61, 39, 49, 98, 42 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Conjectured to be a rearrangement of the natural numbers.

For n>2, a(n) <= a(1)+...+a(n-1). Proof: a(1)+...+a(n-1) >= max { a(i), i=1..n-1}, so a(1)+...+a(n-1) is always a candidate for a(n). QED. So the definition may be changed to: a(1)=1, a(2)=2; for n>2, a(n) is the smallest number not already present which is a divisor of a(1)+...+a(n-1). - N. J. A. Sloane, Nov 05 2005

Except for first two terms, same as A094339. - David Wasserman, Jan 06 2009

A253443(n) = smallest missing number within the first n terms. - Reinhard Zumkeller, Jan 01 2015

LINKS

Richard J. Mathar and Reinhard Zumkeller, Table of n, a(n) for n = 1..10000 (first 789 terms from Richard J. Mathar)

EXAMPLE

Let s(n) = A109735(n) = sum(a(1..n)):

.                   | divisors of s(n),

.                   | in brackets when occurring in a(1..n)

.   n | a(n) | s(n) | A027750(s(n),1..A000005(s(n)))

.  ---+------+------+---------------------------------------------------

.   1 |    1 |    1 | (1)

.   2 |    2 |    3 | (1)  3

.   3 |    3 |    6 | (1 2 3)  6

.   4 |    6 |   12 | (1 2 3)  4  (6)  12

.   5 |    4 |   16 | (1 2 4)  8 16

.   6 |    8 |   24 | (1 2 3 4 6 8)  12 24

.   7 |   12 |   36 | (1 2 3 4 6)  9  (12)  18 36

.   8 |    9 |   45 | (1 3)  5  (9)  15 45

.   9 |    5 |   50 | (1 2 5)  10 25 50

.  10 |   10 |   60 | (1 2 3 4 5 6 10 12)  15 20 30 60

.  11 |   15 |   75 | (1 3 5 15)  25 75

.  12 |   25 |  100 | (1 2 4 5 10)  20  (25)  50 100

.  13 |   20 |  120 | (1 2 3 4 5 6 8 10 12 15 20)  24 30 40 60 120

.  14 |   24 |  144 | (1 2 3 4 6 8 9 12)  16 18  (24)  36 48 72 144

.  15 |   16 |  160 | (1 2 4 5 8 10 16 20)  32 40 80 160

.  16 |   32 |  192 | (1 2 3 4 6 8 12 16 24 32)  48 64 96 192

.  17 |   48 |  240 | (.. 8 10 12 15 16 20 24)  30 40  (48)  60 80 120 240

.  18 |   30 |  270 | (1 2 3 5 6 9 10 15)  18 27  (30)  45 54 90 135 270

.  19 |   18 |  288 | (.. 6 8 9 12 16 18 24 32)  36  (48)  72 96 144 288

.  20 |   36 |  324 | (1 2 3 4 6 9 12 18)  27  (36)  54 81 108 162 324

.  21 |   27 |  351 | (1 3 9)  13  (27)  39 117 351

.  22 |   13 |  364 | (1 2 4)  7  (13)  14 26 28 52 91 182 364

.  23 |    7 |  371 | (1 7)  53 371

.  24 |   53 |  424 | (1 2 4 8 53)  106 212 424

.  25 |  106 |  530 | (1 2 5 10 53 106)  265 530  .

- Reinhard Zumkeller, Jan 05 2015

MAPLE

M:=2000; a:=array(1..M): a[1]:=1: a[2]:=2: as:=convert(a, set): b:=3: for n from 3 to M do t2:=divisors(b) minus as; t4:=sort(convert(t2, list))[1]; a[n]:=t4; b:=b+t4; as:={op(as), t4}; od: aa:=[seq(a[n], n=1..M)]:

MATHEMATICA

a[1] = 1; a[2] = 2; a[n_] := a[n] = Block[{t = Table[a[i], {i, n - 1}]}, s = Plus @@ t; d = Divisors[s]; l = Complement[d, t]; If[l != {}, k = First[l], k = s; While[Position[t, k] == {}, k += s]; k]]; Table[ a[n], {n, 40}] (* Robert G. Wilson v, Aug 12 2005 *)

PROG

(Haskell)

import Data.List (insert)

a109890 n = a109890_list !! (n-1)

a109890_list = 1 : 2 : 3 : f (4, []) 6 where

   f (m, ys) z = g $ dropWhile (< m) $ a027750_row' z where

     g (d:ds) | elem d ys = g ds

              | otherwise = d : f (ins [m, m + 1 ..] (insert d ys)) (z + d)

     ins (u:us) vs'@(v:vs) = if u < v then (u, vs') else ins us vs

-- Reinhard Zumkeller, Jan 02 2015

(Python)

from sympy import divisors

A109890_list, s, y, b = [1, 2], 3, 3, set()

for _ in range(1, 10**3):

....for i in divisors(s):

........if i >= y and not i in b:

............A109890_list.append(i)

............s += i

............b.add(i)

............while y in b:

................b.remove(y)

................y += 1

............break # Chai Wah Wu, Jan 05 2015

CROSSREFS

Cf. A109735, A109736, A111238, A111239, A111240, A111241, A064413 (EKG sequence), A094339, A111315, A111316.

Cf. A027750, A253443, A253444, A095258.

Sequence in context: A112975 A257218 A289055 * A086537 A212486 A127562

Adjacent sequences:  A109887 A109888 A109889 * A109891 A109892 A109893

KEYWORD

easy,nonn,look

AUTHOR

Amarnath Murthy, Jul 13 2005

EXTENSIONS

More terms from Erich Friedman, Aug 08 2005

STATUS

approved

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Last modified March 22 23:04 EDT 2019. Contains 321422 sequences. (Running on oeis4.)