A109890 -id:A109890 - cp's OEIS frontend

A109736 Where n appears in A109890.

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

A111242 Records in A109890.

Original entry on oeis.org

1, 2, 3, 6, 8, 12, 15, 25, 32, 48, 53, 106, 265, 318, 321, 428, 642, 2563, 5713, 15019, 113573, 230801, 306127, 941071, 1016963, 5166859, 9536294, 30956561, 123081011, 265811669, 1016775247, 3050325741, 4354364461, 14086296281, 60060345973

Offset: 1

N. J. A. Sloane, Oct 30 2005

nonn

Cf. A109890, A111243.

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]]; b = 0; t = {}; Do[If[a[n] > b, b = a[n]; AppendTo[t, b]], {n, 2500}]; t (Robert G. Wilson v)

More terms from Robert G. Wilson v, Nov 03 2005

A111243 Where records appear in A109890.

Original entry on oeis.org

1, 2, 3, 4, 6, 7, 11, 12, 16, 17, 24, 25, 26, 28, 32, 34, 35, 52, 116, 154, 222, 232, 256, 279, 504, 568, 598, 634, 844, 909, 1306, 1307, 1505, 1704, 2016, 2050, 2164, 2325, 2497, 2533, 2630, 2631, 2701, 3130, 3225, 3322, 3411, 3524, 3785, 4243, 4244, 4245

Offset: 1

N. J. A. Sloane, Oct 30 2005

nonn

Cf. A109890, A111242.

More terms from David Wasserman, Jan 07 2009

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]]

A253444 Lengths of runs of identical terms in A253443.

Original entry on oeis.org

2, 4, 14, 17, 27, 1, 21, 62, 34, 86, 86, 47, 186, 94, 53, 212, 226, 148, 251, 696, 1484, 630, 1870, 563, 813, 188, 1222, 154, 960, 6654, 1980, 8872, 10027, 3628, 5724, 6330, 12059, 10418, 10169, 4192, 4868

Offset: 1

Reinhard Zumkeller, Jan 01 2015

A253584(n) occurs exactly a(n) times in A253443.

Cf. A253443, A253584, A109890.

import Data.List (group)
a253444 n = a253444_list !! (n-1)
a253444_list = map length $ group a253443_list

nn = 2^14; c[_] := False;
Array[Set[{a[#], c[#]}, {#, True}] &, 2]; u = s = a[1] + a[2];
Differences@ Reap[Monitor[Do[k = SelectFirst[Divisors[s], ! c[#] &];
  c[k] = True; s += k;
If[k == u, Sow[n]; While[c[u], u++]], {n, 3, nn}], n] ][[-1, 1]] (* Michael De Vlieger, Apr 27 2024 *)

More terms from Michael De Vlieger, Apr 27 2024

A253584 Distinct terms in A253443.

Original entry on oeis.org

4, 5, 7, 11, 17, 34, 37, 43, 67, 73, 127, 141, 157, 173, 227, 283, 347, 359, 401, 409, 607, 857, 1091, 1303, 1823, 1907, 2281, 2437, 2441, 2609, 2969, 3851, 4211, 4691, 6907, 7537, 8429, 10301, 11953, 14081, 14557

Offset: 1

Reinhard Zumkeller, Jan 05 2015

a(n) occurs exactly A253444(n) times in A253443.

Cf. A253443, A253444, A109890.

import Data.List (group)
a253584 n = a253584_list !! (n-1)
a253584_list = map head $ group a253443_list

nn = 2^14; c[_] := False;
Array[Set[{a[#], c[#]}, {#, True}] &, 2]; u = s = a[1] + a[2];
Rest@ Reap[Monitor[Do[k = SelectFirst[Divisors[s], ! c[#] &];
  c[k] = True; s += k;
If[k == u, Sow[u]; While[c[u], u++]], {n, 3, nn}], n]][[-1, 1]] (* Michael De Vlieger, Apr 27 2024 *)

More terms from 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.

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.

A372284 Primes in the order in which they appear in A124652.

Original entry on oeis.org

2, 3, 5, 11, 7, 31, 13, 19, 17, 37, 29, 41, 47, 59, 23, 97, 53, 829, 71, 149, 101, 167, 79, 151, 43, 103, 157, 113, 139, 109, 137, 197, 353, 89, 277, 293, 269, 229, 73, 61, 571, 83, 191, 691, 251, 179, 127, 257, 193, 173, 239, 163, 331, 613, 617, 311, 67, 523

Offset: 1

Michael De Vlieger, Apr 29 2024

nonn

Analogous to A111239, a sequence which instead pertains to A109890.

			Let b(x) = A372009(x).
Table of first terms:
   n  b(n) 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

Cf. A124652, A372009.

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[k]]; c[k] = True;
    s += k, {i, 3, nn}] ][[-1, 1]]

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]]

cp's OEIS Frontend

A109736 Where n appears in A109890.

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A111242 Records in A109890.

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A111243 Where records appear in A109890.

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A372111 Partial sums of A124652.

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A253444 Lengths of runs of identical terms in A253443.

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A253584 Distinct terms in A253443.

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

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

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A372284 Primes in the order in which they appear in A124652.

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