A095258 -id:A095258 - cp's OEIS frontend

A095259 Smallest m such that A095258(m) = n.

1, 3, 2, 4, 10, 5, 21, 9, 6, 16, 18, 14, 19, 39, 25, 26, 11, 8, 51, 38, 46, 56, 20, 15, 17, 34, 7, 48, 50, 37, 78, 27, 67, 12, 52, 33, 64, 66, 69, 44, 169, 53, 134, 93, 95, 40, 22, 28, 47, 41, 29, 89, 91, 54, 96, 60, 70, 99, 312, 43, 202, 80, 157, 55, 63, 92, 130, 13

Offset: 1

Reinhard Zumkeller, May 31 2004

nonn

Inverse of A095258, if the conjecture is true, that this is a permutation: a(A095258(n)) = A095258(a(n)) = n;

A095261(n) = a(a(n)).

Index entries for sequences that are permutations of the natural numbers

import Data.List (elemIndex); import Data.Maybe (fromJust)
a095259 = (+ 1) . fromJust . (`elemIndex` a095258_list)
-- Reinhard Zumkeller, Dec 31 2014

Revised version: Reinhard Zumkeller, Dec 31 2014

A253415 Smallest missing number within the first n terms in A095258.

Original entry on oeis.org

2, 4, 5, 5, 5, 5, 5, 5, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 19, 19, 19, 19, 19, 19, 19, 19, 19, 19, 19, 19, 22, 22, 22, 22, 22, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31, 31

Offset: 2

Reinhard Zumkeller, Dec 31 2014

nonn

Michael De Vlieger, Table of n, a(n) for n = 2..10000 (terms 2..797 from Reinhard Zumkeller)

Cf. A095258, A095259, A253425 (run lengths).

import Data.List (delete)
a253415 n = a253415_list !! (n-2)
a253415_list = f [2..] 1 where
   f xs z = g xs where
     g (y:ys) = if mod z' y > 0 then g ys else x : f xs' (z + y)
                where xs'@(x:_) = delete y xs
     z' = z + 2
-- Reinhard Zumkeller, Dec 31 2014

c[] = 0; c[1] = j = 1; u = 2; s = 3; Reap[Do[d = Divisors[s]; k = 1; While[c[d[[k]]] > 0, k++]; Set[k, d[[k]]]; Set[c[k], i]; If[k == u, While[c[u] > 0, u++]]; Sow[u]; j = k; s += k, {i, 2, 2^12}]][[-1, -1]] (* _Michael De Vlieger, Jan 23 2022 *)

from itertools import islice
from sympy import divisors
def A253415_gen(): # generator of terms, first term is a(2)
    bset, m, s = {1}, 2, 3
    while True:
        for d in divisors(s):
            if d not in bset:
                bset.add(d)
                while m in bset:
                    m += 1
                yield m
                s += d
                break
A253415_list = list(islice(A253415_gen(),30)) # Chai Wah Wu, Jan 25 2022

A095260 a(n) = A095258(A095258(n)).

Original entry on oeis.org

1, 2, 3, 4, 9, 8, 32, 11, 18, 6, 25, 26, 99, 34, 235, 5, 15, 17, 68, 94, 27, 49, 241, 243, 24, 10, 102, 28, 19, 10519, 505, 74, 219, 16, 119, 506, 289, 23, 12, 21, 29, 925, 56, 46, 116, 7, 113, 48, 193, 51, 13, 73, 470, 54, 37, 47, 72, 174, 85, 22, 471, 479, 131

Offset: 1

Reinhard Zumkeller, May 31 2004

nonn

Integer permutation with inverse A095261: a(A095261(n)) = A095261(a(n)) = n;

A095259(a(n)) = a(A095259(n)) = A095258(n).

Index entries for sequences that are permutations of the natural numbers

Cf. A095258, A095259, A095261.

A350928 {Partial sums of A095258} + 2.

Original entry on oeis.org

3, 6, 8, 12, 18, 27, 54, 72, 80, 85, 102, 136, 204, 216, 240, 250, 275, 286, 299, 322, 329, 376, 470, 705, 720, 736, 768, 816, 867, 1156, 1734, 1836, 1872, 1898, 1971, 2190, 2220, 2240, 2254, 2300, 2350, 2820, 2880, 2920, 3066, 3087, 3136, 3164, 3277, 3306, 3325

Offset: 1

Michael De Vlieger, Jan 23 2022

Let A095258(n) = m | a(n) such that m is minimal and distinct in A095258.

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Michael De Vlieger, Log-log scatterplot of a(n), b(n), and c(n), n = 1..10^5, where a(n) appears in red, b(n) = A095258(n) in blue, A350929(n) in gold.

Cf. A095258, A350929.

c[_] = 0; j = c[1] = 1; s = 3; Reap[Do[d = Divisors[s]; k = 1; While[c[d[[k]]] > 0, k++]; Set[k, d[[k]]]; Set[c[k], i]; Sow[s]; j = k; s += k, {i, 2, 52}]][[-1, -1]]

a(n) = A095258(n) * A350929(n).

A350929 a(n) = A350928(n)/A095258(n).

Original entry on oeis.org

3, 2, 4, 3, 3, 3, 2, 4, 10, 17, 6, 4, 3, 18, 10, 25, 11, 26, 23, 14, 47, 8, 5, 3, 48, 46, 24, 17, 17, 4, 3, 18, 52, 73, 27, 10, 74, 112, 161, 50, 47, 6, 48, 73, 21, 147, 64, 113, 29, 114, 175, 96, 81, 64, 55, 161, 47, 12, 48, 73, 15, 13, 74, 131, 38, 132, 153

Offset: 1

Michael De Vlieger, Jan 23 2022

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A095258, A350928.

c[_] = 0; j = c[1] = 1; s = 3; Reap[Do[d = Divisors[s]; k = 1; While[c[d[[k]]] > 0, k++]; Set[k, d[[k]]]; Set[c[k], i]; Sow[s/j]; j = k; s += k, {i, 2, 68}]][[-1, -1]]

A308751 a(n) = (2 + Sum_{k = 1..n-1} A095258(k)) / A095258(n).

Original entry on oeis.org

2, 1, 3, 2, 2, 2, 1, 3, 9, 16, 5, 3, 2, 17, 9, 24, 10, 25, 22, 13, 46, 7, 4, 2, 47, 45, 23, 16, 16, 3, 2, 17, 51, 72, 26, 9, 73, 111, 160, 49, 46, 5, 47, 72, 20, 146, 63, 112, 28, 113, 174, 95, 80, 63, 54, 160, 46, 11, 47, 72, 14, 12, 73, 130, 37, 131, 152, 51

Offset: 1

Rémy Sigrist, Jun 22 2019

Are there infinitely many 1's in this sequence?

			a(3) = (2 + A095258(1) + A095258(2)) / A095258(3) = (2 + 1 + 3) / 2 = 3.

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, PARI program for A308751

Cf. A095258.

PARI
```
See Links section.
```

from itertools import islice
from sympy import divisors
def A308751_gen(): # generator of terms
    bset, s = {1}, 3
    yield 2
    while True:
        for d in divisors(s):
            if d not in bset:
                yield s//d
                bset.add(d)
                s += d
                break
A308751_list = list(islice(A308751_gen(),30)) # Chai Wah Wu, Jan 25 2022

A350741 Records in A095258.

Original entry on oeis.org

1, 3, 4, 6, 9, 27, 34, 68, 94, 235, 289, 578, 799, 1921, 9683, 16021, 27421, 54842, 69301, 138602, 434789, 1787371, 5179771, 5655149, 9653251, 10209853, 20419706, 43184409, 301039141, 611363527, 1274384647, 5084853899, 14906805553, 14946637163, 22591381313, 69291164983

Offset: 1

Michael De Vlieger, Jan 23 2022

nonn

Michael De Vlieger, Table of n, a(n) for n = 1..161

Cf. A095258.

c[_] = 0; j = c[1] = r = 1; s = 3; Prepend[Reap[Do[d = Divisors[s]; k = 1; While[c[d[[k]]] > 0, k++]; Set[k, d[[k]]]; Set[c[k], i]; If[k > r, r = k; Sow[r]]; j = k; s += k, {i, 2, 2100}] ][[-1, -1]], 1]

from itertools import islice
from sympy import divisors
def A350741_gen(): # generator of terms
    bset, c, s = {1}, 1, 3
    yield 1
    while True:
        for d in divisors(s):
            if d not in bset:
                if d > c:
                    yield d
                    c = d
                bset.add(d)
                s += d
                break
A350741_list = list(islice(A350741_gen(),20)) # Chai Wah Wu, Jan 25 2022

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

A253443 Smallest missing number within the first n terms in A109890.

Original entry on oeis.org

4, 4, 5, 5, 5, 5, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 34, 37, 37, 37, 37, 37

Offset: 4

Reinhard Zumkeller, Jan 01 2015

nonn

A253584(n) occurs exactly A253444(n) times.

Reinhard Zumkeller, Table of n, a(n) for n = 4..10000

Cf. A095258, A095259, A253444 (run lengths), A253584 (range), A253415.

import Data.List (insert)
a253443 n = a253443_list !! (n-4)
a253443_list = f (4, []) 6 where
   f (m,ys) z = g $ dropWhile (< m) $ a027750_row' z where
     g (d:ds) | elem d ys = g ds
              | otherwise = m : 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 03 2015

A095261 A095259(A095259(n)).

Original entry on oeis.org

1, 2, 3, 4, 16, 10, 46, 6, 5, 26, 8, 39, 51, 69, 17, 34, 18, 9, 29, 66, 40, 60, 38, 25, 11, 12, 21, 28, 41, 64, 112, 7, 130, 14, 89, 67, 55, 92, 71, 93, 448, 91, 189, 107, 85, 44, 56, 48, 22, 169, 50, 273, 158, 54, 76, 43, 133, 68, 417, 134, 239, 100, 288, 96, 157

Offset: 1

Reinhard Zumkeller, May 31 2004

nonn

Integer permutation with inverse A095260: a(A095260(n))=A095260(a(n))=n;

A095258(a(n))=a(A095258(n))=A095259(n).

Index entries for sequences that are permutations of the natural numbers

cp's OEIS Frontend

A095259 Smallest m such that A095258(m) = n.

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A253415 Smallest missing number within the first n terms in A095258.

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A095260 a(n) = A095258(A095258(n)).

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A350928 {Partial sums of A095258} + 2.

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A350929 a(n) = A350928(n)/A095258(n).

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A308751 a(n) = (2 + Sum_{k = 1..n-1} A095258(k)) / A095258(n).

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A350741 Records in A095258.

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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).

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A253443 Smallest missing number within the first n terms in A109890.