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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, 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) . 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 Cf. A109735, A109736, A111238, A111239, A111240, A111241, A064413 (EKG sequence), A094339, A111315, A111316. Cf. A027750, A253443, A253444, A095258. Sequence in context: A257218 A349702 A289055 * A370046 A373326 A086537 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 July 22 19:00 EDT 2024. Contains 374540 sequences. (Running on oeis4.)