%I #41 Apr 20 2024 11:00:46
%S 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,
%T 265,159,318,212,14,107,321,214,428,642,535,35,21,181,11,33,22,23,59,
%U 70,28,151,29,19,233,466,2563,699,932,40,26,38,31,61,39,49,98,42
%N 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).
%C Conjectured to be a rearrangement of the natural numbers.
%C 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
%C Except for first two terms, same as A094339. - _David Wasserman_, Jan 06 2009
%C A253443(n) = smallest missing number within the first n terms. - _Reinhard Zumkeller_, Jan 01 2015
%H Richard J. Mathar and Reinhard Zumkeller, <a href="/A109890/b109890.txt">Table of n, a(n) for n = 1..10000</a> (first 789 terms from Richard J. Mathar)
%H Michael De Vlieger, <a href="/A109890/a109890.txt">Mathematica algorithm for this sequence and A109735 that avoids searching lists to speed output</a>
%H Michael De Vlieger, <a href="/A109890/a109890.png">Log log scatterplot of a(n)</a>, n = 1..2^16, showing primes in red, perfect powers of primes in gold, squarefree composites in green, and other numbers in blue.
%e Let s(n) = A109735(n) = sum(a(1..n)):
%e . | divisors of s(n),
%e . | in brackets when occurring in a(1..n)
%e . n | a(n) | s(n) | A027750(s(n),1..A000005(s(n)))
%e . ---+------+------+---------------------------------------------------
%e . 1 | 1 | 1 | (1)
%e . 2 | 2 | 3 | (1) 3
%e . 3 | 3 | 6 | (1 2 3) 6
%e . 4 | 6 | 12 | (1 2 3) 4 (6) 12
%e . 5 | 4 | 16 | (1 2 4) 8 16
%e . 6 | 8 | 24 | (1 2 3 4 6 8) 12 24
%e . 7 | 12 | 36 | (1 2 3 4 6) 9 (12) 18 36
%e . 8 | 9 | 45 | (1 3) 5 (9) 15 45
%e . 9 | 5 | 50 | (1 2 5) 10 25 50
%e . 10 | 10 | 60 | (1 2 3 4 5 6 10 12) 15 20 30 60
%e . 11 | 15 | 75 | (1 3 5 15) 25 75
%e . 12 | 25 | 100 | (1 2 4 5 10) 20 (25) 50 100
%e . 13 | 20 | 120 | (1 2 3 4 5 6 8 10 12 15 20) 24 30 40 60 120
%e . 14 | 24 | 144 | (1 2 3 4 6 8 9 12) 16 18 (24) 36 48 72 144
%e . 15 | 16 | 160 | (1 2 4 5 8 10 16 20) 32 40 80 160
%e . 16 | 32 | 192 | (1 2 3 4 6 8 12 16 24 32) 48 64 96 192
%e . 17 | 48 | 240 | (.. 8 10 12 15 16 20 24) 30 40 (48) 60 80 120 240
%e . 18 | 30 | 270 | (1 2 3 5 6 9 10 15) 18 27 (30) 45 54 90 135 270
%e . 19 | 18 | 288 | (.. 6 8 9 12 16 18 24 32) 36 (48) 72 96 144 288
%e . 20 | 36 | 324 | (1 2 3 4 6 9 12 18) 27 (36) 54 81 108 162 324
%e . 21 | 27 | 351 | (1 3 9) 13 (27) 39 117 351
%e . 22 | 13 | 364 | (1 2 4) 7 (13) 14 26 28 52 91 182 364
%e . 23 | 7 | 371 | (1 7) 53 371
%e . 24 | 53 | 424 | (1 2 4 8 53) 106 212 424
%e . 25 | 106 | 530 | (1 2 5 10 53 106) 265 530 .
%e - _Reinhard Zumkeller_, Jan 05 2015
%p 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)]:
%t 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 *)
%o (Haskell)
%o import Data.List (insert)
%o a109890 n = a109890_list !! (n-1)
%o a109890_list = 1 : 2 : 3 : f (4, []) 6 where
%o f (m,ys) z = g $ dropWhile (< m) $ a027750_row' z where
%o g (d:ds) | elem d ys = g ds
%o | otherwise = d : f (ins [m, m + 1 ..] (insert d ys)) (z + d)
%o ins (u:us) vs'@(v:vs) = if u < v then (u, vs') else ins us vs
%o -- _Reinhard Zumkeller_, Jan 02 2015
%o (Python)
%o from sympy import divisors
%o A109890_list, s, y, b = [1, 2], 3, 3, set()
%o for _ in range(1,10**3):
%o for i in divisors(s):
%o if i >= y and i not in b:
%o A109890_list.append(i)
%o s += i
%o b.add(i)
%o while y in b:
%o b.remove(y)
%o y += 1
%o break # _Chai Wah Wu_, Jan 05 2015
%Y Cf. A109735, A109736, A111238, A111239, A111240, A111241, A064413 (EKG sequence), A094339, A111315, A111316.
%Y Cf. A027750, A253443, A253444, A095258.
%K easy,nonn,look
%O 1,2
%A _Amarnath Murthy_, Jul 13 2005
%E More terms from _Erich Friedman_, Aug 08 2005