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)
Michael De Vlieger, Mathematica algorithm for this sequence and A109735 that avoids searching lists to speed output
Michael De Vlieger, Log log scatterplot of a(n), n = 1..2^16, showing primes in red, perfect powers of primes in gold, squarefree composites in green, and other numbers in blue.
EXAMPLE
Let s(n) = A109735(n) = sum(a(1..n)):
. | divisors of s(n),
. | in brackets when occurring in a(1..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 i not 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
KEYWORD
AUTHOR
Amarnath Murthy, Jul 13 2005
EXTENSIONS
More terms from Erich Friedman, Aug 08 2005
STATUS
approved