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).
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
Offset: 1
Examples
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
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.
Crossrefs
Programs
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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 -
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 *) -
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
Extensions
More terms from Erich Friedman, Aug 08 2005
Comments