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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). 22
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)
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)
. 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 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
Sequence in context: A257218 A349702 A289055 * A370046 A373326 A086537
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 July 22 19:00 EDT 2024. Contains 374540 sequences. (Running on oeis4.)