A111316 -id:A111316 - cp's OEIS frontend

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

Amarnath Murthy, Jul 13 2005

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

			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

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.

Cf. A109735, A109736, A111238, A111239, A111240, A111241, A064413 (EKG sequence), A094339, A111315, A111316.

Cf. A027750, A253443, A253444, A095258.

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

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)]:

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 *)

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

More terms from Erich Friedman, Aug 08 2005

A111315 Positions where A109890(n) = Sum_{i = 1..n-1} A109890(i).

Original entry on oeis.org

3, 4, 222, 232, 279, 568, 634, 844, 1307, 1704, 2016, 2050, 2164, 2325, 2701, 3130, 3225, 3322, 3524, 4673, 6007, 6692, 7141, 7221, 7954, 8689, 11051, 11898, 13959, 14166, 16192, 17078, 17220, 17521, 18071, 22324, 22414, 24757, 28290, 28979

Offset: 1

N. J. A. Sloane, Nov 04 2005

nonn

These are the positions where A109735(n) = 2*A109735(n-1), or equally, where A111241(n) = 1.

Cf. A109890, A109735, A111316, A111241.

s = 3; S = Set([1, 2]); i = 3; while (1, d = divisors(s); j = 1; while (setsearch(S, d[j]), j++); n = d[j]; if (n == s, print(i)); s += n; S = setunion(S, Set(n)); i++); \\ David Wasserman, Jan 09 2009

More terms from David Wasserman, Jan 09 2009

A111241 a(n) = A109735(n)/A109890(n+1).

Original entry on oeis.org

1, 1, 3, 2, 2, 4, 9, 5, 4, 3, 5, 5, 9, 5, 4, 8, 15, 8, 12, 27, 52, 7, 4, 2, 5, 3, 6, 106, 14, 5, 9, 5, 4, 6, 107, 180, 21, 362, 121, 183, 176, 69, 59, 150, 28, 151, 232, 19, 10, 2, 11, 9, 233, 360, 247, 304, 155, 244, 195, 98, 231, 174, 196, 50, 591, 296, 198, 51, 199

Offset: 2

N. J. A. Sloane, Oct 30 2005

nonn

This is always an integer for n>=2.

a(n) = 1 for n in A111315. When this happens A109890(n+1) makes a large jump. The corresponding values of A109890(n+1) are in A111316 (cf. A111242).

			A109735(4)=12, A109890(5)=4, so a(4) = 12/4 = 3.

Michael De Vlieger, Table of n, a(n) for n = 2..10000

Cf. A109735, A109890, A094340, A111315, A111316, A111242.

Different from A094339.

nn = 71; c[_] := False;
Array[Set[{a[#], c[#]}, {#, True}] &, 2]; s = a[1] + a[2];
Reap[Do[k = SelectFirst[Divisors[s], ! c[#] &];
    c[k] = True; Sow[s/k];
s += k, {n, 3, nn}] ][[-1, 1]] (* Michael De Vlieger, Apr 26 2024 *)

a(n) = A094340(n) for all n > 1. - David Wasserman, Jan 06 2009

A371910 Position of A109890(n) among the sorted set of divisors of A109735(n-1).

Original entry on oeis.org

2, 4, 4, 4, 7, 6, 3, 4, 9, 5, 6, 12, 9, 9, 11, 14, 9, 13, 9, 4, 4, 3, 6, 7, 6, 10, 12, 5, 5, 6, 8, 9, 13, 12, 4, 15, 5, 3, 4, 6, 8, 4, 9, 17, 7, 2, 5, 3, 8, 7, 6, 13, 8, 17, 6, 7, 4, 9, 10, 8, 13, 17, 15, 7, 3, 7, 13, 5, 6, 16, 8, 11, 8, 5, 4, 13, 12, 17, 5, 6

Offset: 3

Michael De Vlieger, Apr 26 2024

nonn

A109890(n) is the a(n)-th smallest divisor of A109735(n-1).

			Table relating sequences b = A109890, s = A109735, c = A371909. a(n) = c(n) implies both A111315(i) = n and A111316(i) = b(n) = s(n-1).
    n     b(n)   s(n-1)  a(n)  c(n)    i
   --------------------------------------
    3       3   =    3     2     2     1
    4       6   =    6     4     4     2
    5       4       12     4     6
    6       8       16     4     5
    7      12       24     7     8
    8       9       36     6     9
    9       5       45     3     6
   10      10       50     4     6
   11      15       60     9    12
   12      25       75     5     6
  ...
  222  113573 = 113573     4     4     3
  ...
  232  230801 = 230801     4     4     4
  ...
  279  941071 = 941071     4     4     5
  ...

Michael De Vlieger, Table of n, a(n) for n = 3..10000

Cf. A000005, A027750, A109735, A109890, A111315, A111316, A371909.

nn = 120; c[_] := False;
Array[Set[{a[#], c[#]}, {#, True}] &, 2]; s = a[1] + a[2];
Reap[Do[d = Divisors[s]; k = SelectFirst[d, ! c[#] &];
    c[k] = True; Sow[FirstPosition[d, k][[1]]];
    s += k, {n, 3, nn}] ][[-1, 1]]

1 < a(n) <= A371909(n), where A371909(n) = A000005(A109735(n-1)), corollary of Sloane's theorem in the comments in A109890.
A109890(n) = T(j, k), where T = A027750, j = A109735(n-1), and k = a(n).
A371909(n) = A371910(n) if and only if A109890(n) = A109735(n-1).

cp's OEIS Frontend

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).

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A111315 Positions where A109890(n) = Sum_{i = 1..n-1} A109890(i).

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A111241 a(n) = A109735(n)/A109890(n+1).

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A371910 Position of A109890(n) among the sorted set of divisors of A109735(n-1).

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