A109735 -id:A109735 - cp's OEIS frontend

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

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).

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

A372111 Partial sums of A124652.

Original entry on oeis.org

1, 3, 6, 10, 15, 24, 30, 38, 54, 66, 77, 84, 98, 126, 144, 168, 189, 216, 248, 279, 360, 370, 390, 403, 572, 594, 627, 646, 663, 702, 728, 777, 814, 858, 894, 942, 996, 1060, 1085, 1120, 1160, 1189, 1230, 1245, 1290, 1320, 1370, 1450, 1508, 1560, 1620, 1692, 1739

Offset: 1

Michael De Vlieger, Apr 25 2024

nonn

Analogous to A109735 with respect to A109890.

A007947(A124652(n+1)) | a(n) for n > 2.

			See A124652.

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

Cf. A007947, A109890, A109735, A124652.

nn = 54; c[_] := False; a[1] = 1; a[2] = 2; s = u = 3; c[1] = c[2] = True;
f[x_] := f[x] = Times @@ FactorInteger[x][[All, 1]];
{1}~Join~Reap[Do[Sow[s]; k = u; While[Or[Mod[s, f[k]] != 0, c[k]], k++];
    Set[{a[n], c[k]}, {k, True}];
    s += k; If[k == u, While[c[u], u++]], {n, 3, nn}] ][[-1, 1]]

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

A371909 Number of divisors of the partial sums of A109890.

Original entry on oeis.org

1, 2, 4, 6, 5, 8, 9, 6, 6, 12, 6, 9, 16, 15, 12, 14, 20, 16, 18, 15, 8, 12, 4, 8, 8, 8, 12, 16, 12, 8, 8, 12, 12, 16, 16, 8, 48, 8, 8, 8, 16, 20, 8, 16, 48, 12, 4, 16, 4, 12, 8, 8, 18, 16, 60, 12, 20, 8, 24, 24, 15, 36, 40, 36, 12, 8, 20, 32, 8, 12, 36, 16, 24

Offset: 1

Michael De Vlieger, Apr 26 2024

nonn

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

Cf. A000005, A109735, A109890, A371910.

nn = 120; c[_] := False;
Array[Set[{a[#], c[#]}, {#, True}] &, 2]; s = a[1] + a[2];
{1}~Join~Reap[Do[k = SelectFirst[Divisors[s], ! c[#] &];
     c[k] = True; Sow[DivisorSigma[0, s]];
     s += k, {n, 3, nn}] ][[-1, 1]]

a(n) = A000005(A109735(n)).

A372028 Numbers k such that A124652(k) divides A372111(k-1).

Original entry on oeis.org

3, 5, 7, 11, 12, 13, 15, 16, 17, 18, 20, 22, 24, 26, 27, 28, 29, 30, 31, 33, 40, 41, 42, 43, 44, 46, 49, 50, 51, 53, 55, 58, 59, 60, 62, 63, 64, 66, 67, 68, 69, 70, 71, 72, 73, 78, 79, 80, 92, 93, 95, 98, 101, 102, 103, 104, 105, 107, 109, 111, 112, 115, 116, 117

Offset: 1

Michael De Vlieger, May 05 2024

nonn

Contains A372009(m), m > 1.

For k in this sequence, A124652(k) has the same relationship with A372111(k-1) as A109890(i) has with A109735(i-1) for i > 2.

			Let b(x) = A124652(x) and s(x) = A372111(x), where A372111 contains partial sums of A124652.
a(1) = 3 since b(3) = 3, a divisor of s(2) = 3.
a(2) = 5 since b(5) = 5, a divisor of s(4) = 10.
a(3) = 7 since b(7) = 6, a divisor of s(6) = 24, etc.

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Michael De Vlieger, Log log scatterplot of A124652(n), n = 1..10^5, showing A124652(a(n)) in green, but A124652(a(n)) that are prime in red.

Cf. A007947, A027750, A109735, A109890, A124652, A372009, A372111, A372399.

nn = 120; 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];
Reap[Do[r = f[s]; k = SelectFirst[r, ! c[#] &];
  If[Divisible[s, k], Sow[i]]; c[k] = True;
  s += k, {i, 3, nn}] ][[-1, 1]]

A124652(a(n)) is a number in row A372111(a(n)-1) of A027750.

A372322 a(n) = A010846(A372111(n)).

Original entry on oeis.org

1, 2, 5, 6, 5, 11, 18, 8, 16, 22, 5, 28, 13, 33, 23, 38, 11, 26, 12, 9, 58, 28, 80, 5, 30, 55, 19, 27, 19, 56, 37, 21, 27, 87, 44, 44, 48, 38, 18, 58, 42, 5, 110, 26, 112, 140, 38, 45, 32, 144, 102, 59, 5, 139, 225, 39, 44, 22, 180, 86, 114, 34, 23, 133, 41, 115

Offset: 1

Michael De Vlieger, May 05 2024

nonn

Let r(x) = A010846(x), the number of m <= x such that rad(m) | x, where rad = A007947.
Let row k of A162306 contain { m : rad(m) | k, m <= k }. Thus r(k) is the length of row k of A162306.
a(n) is the length of row A372111(n) of A162306.
Analogous to A371909, which instead regards A109890 and A109735.

			Let s(x) = A372111(x) and let r(x) = A010846(x).
a(1) = 1 since r(s(1)) = r(1) = 1.
a(2) = 2 since r(s(2)) = r(3) = 2. For prime p, r(p) = card({1, p}) = 2.
a(3) = 5 since r(s(3)) = r(6) = 5. r(6) = card({1, 2, 3, 4, 6}) = 5.
a(4) = 6 since r(s(4)) = r(10) = 6. r(10) = card({1, 2, 4, 5, 8, 10}) = 6.
a(5) = 5 since r(s(5)) = r(15) = 5. r(15) = card({1, 3, 5, 9, 15}) = 5.
a(6) = 11 since r(s(6)) = r(24) = 11. r(24) = card({1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24}) = 11, etc.

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

Cf. A007947, A010846, A124652, A162306, A371909, A372111, A372323.

nn = 68; 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];
{1}~Join~Reap[Do[r = f[s]; k = SelectFirst[r, ! c[#] &];
  Sow[Length[r]]; c[k] = True;
  s += k, {i, 3, nn}] ][[-1, 1]]

A372323 A124652(n) is the a(n)-th term in row A372111(n-1) of irregular triangle A162306.

Original entry on oeis.org

2, 4, 4, 4, 5, 7, 5, 8, 8, 2, 10, 8, 12, 11, 13, 6, 13, 6, 6, 9, 8, 11, 4, 8, 16, 5, 6, 7, 13, 12, 7, 10, 19, 15, 16, 17, 9, 6, 15, 10, 3, 11, 8, 18, 28, 14, 14, 10, 30, 28, 15, 4, 20, 33, 13, 12, 6, 22, 18, 21, 12, 11, 29, 12, 11, 8, 24, 18, 8, 14, 17, 32, 33

Offset: 3

Michael De Vlieger, May 05 2024

nonn

Let b(x) = A124652(x) and let s(x) = A372111(x), where A372111 contains partial sums of A124652.
Let r(x) = A010846(x), the number of m <= x such that rad(m) | x, where rad = A007947.
Let row k of A162306 contain { m : rad(m) | k, m <= k }. Thus r(k) is the length of row k of A162306.
Let T(k,j) represent the j-th term in row k of irregular triangle A162306.
a(n) = j is the position of b(n) in row s(n-1) of A162306.
b(n) = T(s(n-1), a(n)).
Analogous to A371910, which instead regards A109890 and A109735.

			Let b(x) = A124652(x) and let s(x) = A372111(x), where A372111 contains partial sums of A124652.
a(3) = 2 since b(3) = 3 is the 2nd term in row s(3) = 3 of A162306, {1, [3]}.
a(4) = 4 since b(4) = 4 is the 4th term in row s(4) = 6 of A162306, {1, 2, 3, [4], 6}.
a(5) = 4 since b(5) = 5 is T(s(n-1), 4) = T(10, 4), {1, 2, 4, [5], 8, 10}.
a(6) = 4 since b(6) = 9 is T(s(n-1), 4) = T(15, 4), {1, 3, 5, [9], 15}.
a(7) = 5 since b(7) = 6 is T(s(n-1), 5) = T(24, 5), {1, 2, 3, 4, [6], 8, 9, 12, 16, 18, 24}, etc.
Table relating this sequence to b = A124652, s = A372111, r = A372322, and A162306.
   n b(n) s(n-1) a(n) r(n) row s(n-1) of A162306
  ---------------------------------------------------------------------
   3    3    3    2    2   {1, [3]}
   4    4    6    4    5   {1, 2, 3, [4], 6}
   5    5   10    4    6   {1, 2, 4, [5], 8, 10}
   6    9   15    4    5   {1, 3, 5, [9], 15}
   7    6   24    5   11   {1, 2, 3, 4, [6], ..., 24}
   8    8   30    7   18   {1, 2, 3, 4, 5, 6, [8], ..., 30}
   9   16   38    5    8   {1, 2, 4, 8, [16], 19, 32, 38}
  10   12   54    8   16   {1, 2, 3, 4, 6, 8, 9, [12], ..., 54}
  11   11   66    8   22   {1, 2, 3, 4, 6, 8, 9, [11], ..., 66}
  12    7   77    2    5   {1, [7], 11, 49, 77}
  13   14   84   10   28   {1, 2, 3, 4, ..., 12, [14], ..., 84}
  14   28   98    8   13   {1, 2, 4, 7, ..., 16, [28], ..., 98}

Michael De Vlieger, Table of n, a(n) for n = 3..10000
Michael De Vlieger, Bar chart showing a(n)/A372322(n-1) for n = 3..1024. This chart illustrates the "depth" of A124652(n) among the terms of the A372111(n-1)-th row of A162306.

Cf. A007947, A010846, A124652, A162306, A371910, A372111, A372322.

nn = 75; 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];
Reap[Do[r = f[s]; k = SelectFirst[r, ! c[#] &];
  Sow[FirstPosition[r, k][[1]]]; c[k] = True;
  s += k, {i, 3, nn}] ][[-1, 1]]

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

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

A372111 Partial sums of A124652.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

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

Views

Author

Keywords

Comments

Crossrefs

Programs

PARI

Extensions

A371909 Number of divisors of the partial sums of A109890.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

A372028 Numbers k such that A124652(k) divides A372111(k-1).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A372322 a(n) = A010846(A372111(n)).

Views

Author

Keywords