A111238 -id:A111238 - 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

A111239 Primes in the order in which they appear in A109890.

Original entry on oeis.org

2, 3, 5, 13, 7, 53, 107, 181, 11, 23, 59, 151, 29, 19, 233, 31, 61, 197, 17, 199, 41, 193, 97, 109, 37, 281, 47, 71, 131, 79, 149, 103, 241, 137, 191, 239, 113, 163, 43, 653, 617, 853, 673, 89, 937, 67, 571, 599, 751, 83, 101, 1103, 829, 457, 499, 229

Offset: 1

N. J. A. Sloane, Oct 30 2005

nonn

Smallest missing prime in A109890 for n <= 10^5 is prime(1821) = 15619. - Michael De Vlieger, Apr 27 2024

Michael De Vlieger, Table of n, a(n) for n = 1..113

Cf. A109890, A111238, A111240.

nn = 2^14; c[_] := False;
Array[Set[{a[#], c[#]}, {#, True}] &, 2];
s = a[1] + a[2]; v = NextPrime[a[2]];
t = Join[{{2, 2}},
  Reap[Monitor[Do[k = SelectFirst[Divisors[s], ! c[#] &];
  c[k] = True; s += k;
  If[PrimeQ[k], Sow[{k, n}];
    If[k == v, While[c[v], v = NextPrime[v]]]], {n, 3, nn}], n] ][[-1, 1]] ];
TakeWhile[t, First[#] <= v &][[All, 1]] (* Michael De Vlieger, Apr 27 2024 *)

A372009 Indices k such that A124652(k) is prime.

Original entry on oeis.org

2, 3, 5, 11, 12, 20, 24, 28, 29, 33, 42, 43, 53, 58, 67, 78, 93, 98, 104, 105, 109, 112, 118, 125, 126, 137, 141, 145, 146, 162, 174, 182, 185, 187, 188, 195, 200, 223, 224, 231, 232, 239, 246, 249, 252, 255, 259, 264, 271, 275, 283, 286, 287, 296, 298, 300, 326

Offset: 1

Michael De Vlieger, Apr 29 2024

nonn

Analogous to A111238, a sequence which instead pertains to A109890.

			Let b(x) = A124652(x).
Table of first terms.
   n  a(n)  b(a(n))
  -----------------
   1    2      2
   2    3      3
   3    5      5
   4   11     11
   5   12      7
   6   20     31
   7   24     13
   8   28     19
   9   29     17
  10   33     37
  11   42     29
  12   43     41
  ...

Michael De Vlieger, Table of n, a(n) for n = 1..2500
Michael De Vlieger, Log log scatterplot of A124652(n), n = 1..25000, showing primes in red.

Cf. A124652, A372028, A372284.

nn = 300; c[_] := False;
rad[x_] := rad[x] = Times @@ FactorInteger[x][[All, 1]];
f[x_] := Select[Range[x], Divisible[x, rad[#]] &];
Array[Set[{a[#], c[#]}, {#, True}] &, 2]; s = a[1] + a[2];
{2}~Join~Reap[Do[r = f[s]; k = SelectFirst[r, ! c[#] &];
    If[PrimeQ[k], Sow[i]]; c[k] = True;
    s += k, {i, 3, nn}] ][[-1, 1]]

Proper subset of A372028.

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).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Maple

Mathematica

Python

Extensions

A111239 Primes in the order in which they appear in A109890.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

A372009 Indices k such that A124652(k) is prime.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula