A094339 -id:A094339 - cp's OEIS frontend

A094340 a(n) = n-th partial sum of A094339 divided by A094339(n+1).

2, 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, 160, 115

Offset: 1

Amarnath Murthy, May 17 2004

nonn

Conjecture: Every natural number occurs in this sequence.

Cf. A094339, A094341.

A094339 := proc(nmax) local a,n,sprev,i; a := [2] ; while nops(a) < nmax do sprev := add(i,i=a) ; n := 1 ; while sprev mod n <> 0 or n in a do n := n+1 ; od ; a := [op(a),n] ; od ; RETURN(a) ; end: A094340 := proc(a094339,n) add( op(i,a094339),i=1..n)/op(n+1,a094339) ; end: a094339 := A094339(100) ; for n from 1 to nops(a094339)-1 do printf("%d, ", A094340(a094339,n)) ; od ; # R. J. Mathar, Apr 30 2007

Corrected and extended by R. J. Mathar, Apr 30 2007

A094341 Index of the occurrence of n in A094339.

Original entry on oeis.org

2, 1, 3, 5, 9, 4, 23, 6, 8, 10, 40, 7, 22, 30, 11, 15, 67, 19, 49, 13, 38, 42, 43, 14, 12, 56, 21, 46, 48, 18, 58, 16, 41, 68, 37, 20, 89, 57, 60, 55, 76, 63, 151, 78, 107, 96, 98, 17, 61, 65, 69, 71, 24, 103, 87, 64, 80, 74, 44, 83, 59, 92, 101, 94, 72, 91, 185, 142, 104, 45

Offset: 1

Amarnath Murthy, May 17 2004

nonn

Cf. A094339, A094340.

A094339 := proc(nmax) local a,n,sprev,i; a := [2] ; while nops(a) < nmax do sprev := add(i,i=a) ; n := 1 ; while sprev mod n <> 0 or n in a do n := n+1 ; od ; a := [op(a),n] ; od ; RETURN(a) ; end: a094339 := A094339(300) : n := 1 : while member(n,a094339,'w') do printf("%d, ",w) ; n := n+1 ; od : # R. J. Mathar, Apr 30 2007

nmax = 70; s = {2};
Do[AppendTo[s, Min[Select[Divisors[Total[s]], !MemberQ[s, #] &]]], {t, 2, 3 nmax}];
a[n_] := FirstPosition[s, n][[1]];
Table[a[n], {n, 1, nmax}] (* Jean-François Alcover, Oct 24 2023, after Ivan Neretin in A094339 *)

Corrected and extended by R. J. Mathar, Apr 30 2007

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).

Original entry on oeis.org

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

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

A109736 Where n appears in A109890.

Original entry on oeis.org

1, 2, 3, 5, 9, 4, 23, 6, 8, 10, 40, 7, 22, 30, 11, 15, 67, 19, 49, 13, 38, 42, 43, 14, 12, 56, 21, 46, 48, 18, 58, 16, 41, 68, 37, 20, 89, 57, 60, 55, 76, 63, 151, 78, 107, 96, 98, 17, 61, 65, 69, 71, 24, 103, 87, 64, 80, 74, 44, 83, 59, 92, 101, 94, 72, 91, 185, 142, 104, 45

Offset: 1

N. J. A. Sloane and Nadia Heninger, Aug 11 2005

nonn

a(10^n): 1, 10, 128, 1430, ... - Robert G. Wilson v, Aug 12 2005
a(n) = A094341(n) for 3 <= n <= 70. - Georg Fischer, Nov 02 2018
According to the remarks in A109890, A094339 and A109890 are essentially the same, just swapping the first 2 terms, so this here is a(n)=A094341(n) for n>=3. - R. J. Mathar, Jul 02 2025

Richard J. Mathar and Reinhard Zumkeller, Table of n, a(n) for n = 1..10000 (first 282 terms from Richard J. Mathar)

Cf. A094341, A095259, A109890.

import Data.List (elemIndex); import Data.Maybe (fromJust)
a109736 = (+ 1) . fromJust . (`elemIndex` a109890_list)
-- Reinhard Zumkeller, Jan 01 2015

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]]; t = Table[a[n], {n, 250}]; Table[k = 1; While[ t[[k]] != n, k++ ]; k, {n, 70}] (* Robert G. Wilson v, Aug 12 2005 *)

More terms from Robert G. Wilson v, Aug 12 2005

cp's OEIS Frontend

A094340 a(n) = n-th partial sum of A094339 divided by A094339(n+1).

Views

Author

Keywords

Comments

Crossrefs

Programs

Maple

Extensions

A094341 Index of the occurrence of n in A094339.

Views

Author

Keywords

Crossrefs

Programs

Maple

Mathematica

Extensions

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

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A109736 Where n appears in A109890.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Haskell

Mathematica

Extensions