A030717 -id:A030717 - cp's OEIS frontend

A030719 Number of new terms at stage n in the formation of A030717.

1, 1, 2, 2, 3, 3, 4, 5, 6, 7, 7, 8, 10, 12, 13, 16, 18, 20, 23, 25, 28, 31, 33, 36, 38, 41, 43, 45, 47, 50, 53, 56, 60, 62, 65, 69, 72, 78, 83, 86, 88, 92, 95, 99, 105, 110, 114, 117, 122, 125, 130, 136, 140, 144, 149, 153, 159, 162, 167, 173, 180, 184, 188

Offset: 1

Clark Kimberling

nonn

Michael De Vlieger, Table of n, a(n) for n = 1..1024 (First 500 terms from Reinhard Zumkeller)

Cf. A030717, A253170.

a030719 n = a030719_list !! (n-1)
a030719_list = tail $ zipWith (-) (tail a253170_list) a253170_list
-- Reinhard Zumkeller, Dec 28 2014

nn = 63; c[] = 0; r = k = a[1] = c[1] = 1; Reap[Do[w = Union@ Array[a, k]; Do[Set[a[j + k], c[w[[j]]]], {j, Length[w]}]; Do[c[a[j + k]]++, {j, Length[w]}]; k += Sow[Length[w]], {nn}]][[-1, -1]] (* _Michael De Vlieger, Nov 15 2022 *)

A030723 Numbers k such that A030717(k) = 1.

Original entry on oeis.org

1, 2, 5, 10, 17, 22, 28, 35, 59, 60, 69, 70, 71, 72, 81, 84, 85, 94, 97, 98, 99, 100, 101, 110, 113, 115, 116, 117, 118, 119, 128, 131, 133, 134, 135, 136, 137, 139, 148, 151, 153, 154, 155, 156, 157, 158, 160, 161, 162, 171, 174

Offset: 1

Clark Kimberling

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..10000
Michael De Vlieger, Plot a(w(j) + k - 1) at (j,k) for j = 1..512 and w the sequence of partial sums of A030719, showing a(m) = 1 in red and a(m) > 1 in light blue.

Cf. A030717, A030724, A030725, A030726.

nn = 102; c[] = 0; m = k = a[1] = c[1] = 1; {1}~Join~Reap[Do[w = Union@ Array[a, k]; Do[If[Set[a[j + k], c[w[[j]]]] == 1, Sow[j + k]], {j, Length[w]}]; Do[c[a[j + k]]++, {j, Length[w]}]; k += Length[w], {nn}]][[-1, -1]] (* _Michael De Vlieger, Nov 15 2022 *)

A030725 Numbers k such that A030717(k) = 3.

Original entry on oeis.org

6, 8, 9, 12, 13, 15, 19, 32, 39, 41, 42, 46, 48, 49, 50, 54, 56, 57, 66, 67, 77, 78, 79, 91, 92, 107, 108, 125, 126, 146, 149, 169, 172, 173, 182, 194, 197, 198, 207, 225, 226, 236, 256, 257, 267, 289, 292, 300, 325, 327, 336, 362

Offset: 1

Clark Kimberling

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..1000
Michael De Vlieger, Plot a(s(j) + k - 1) at (j,k) for j = 1..512 and s = partial sums of A030719, showing a(m) = 3 in red, a(m) = 1 in medium blue for reference, and other values of a(m) in light blue.

Cf. A030723, A030724, A030726.

Cf. A030717.

nn = 30; m = 3; c[] = 0; k = a[1] = c[1] = 1; Reap[Do[w = Union@ Array[a, k]; Do[Set[a[j + k], c[w[[j]]]]; If[a[j + k] == m, Sow[m]], {j, Length[w]}]; Do[c[a[j + k]]++, {j, Length[w]}]; k += Length[w], {n, nn}] ][[-1, -1]] (* _Michael De Vlieger, Nov 19 2022 *)

A030726 Numbers k such that A030717(k) = 4.

Original entry on oeis.org

11, 14, 24, 58, 64, 145, 222, 253, 290, 326, 328, 364, 449, 451, 491, 496, 538, 539, 588, 589, 595, 641, 642, 648, 650, 653, 697, 698, 704, 706, 710, 757, 758, 764, 766, 819, 820, 828, 884, 885, 953, 954, 964, 1025, 1026, 1036, 1103, 1104, 1114, 1186, 1187

Offset: 1

Clark Kimberling

nonn

Michael De Vlieger, Table of n, a(n) for n = 1..10000 (first 1000 terms by Seiichi Manyama)
Michael De Vlieger, Plot a(s(j) + k - 1) at (j,k) for j = 1..512 and s = partial sums of A030719, showing a(m) = 4 in red, a(m) = 1 in medium blue for reference, and other values of a(m) in light blue.

Cf. A030723, A030724, A030725.

Cf. A030717.

nn = 40; m = 4; c[] = 0; k = a[1] = c[1] = 1; Reap[Do[w = Union@ Array[a, k]; Do[Set[a[j + k], c[w[[j]]]]; If[a[j + k] == m, Sow[m]], {j, Length[w]}]; Do[c[a[j + k]]++, {j, Length[w]}]; k += Length[w], {n, nn}] ][[-1, -1]] (* _Michael De Vlieger, Nov 19 2022 *)

More terms from Sean A. Irvine, Apr 08 2020

A030724 Numbers k such that A030717(k) = 2.

Original entry on oeis.org

3, 4, 7, 21, 26, 27, 33, 34, 40, 47, 55, 65, 82, 83, 95, 96, 111, 112, 114, 129, 130, 132, 138, 150, 152, 159, 175, 199, 200, 227, 229, 239, 255, 258, 259, 260, 270, 271, 288, 291, 293, 303, 304, 324, 329, 339, 340, 345, 377, 378

Offset: 1

Clark Kimberling

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..10000
Michael De Vlieger, Plot a(s(j) + k - 1) at (j,k) for j = 1..512 and s = partial sums of A030719, showing a(m) = 2 in red and a(m) != 2 in light blue.

Cf. A030723, A030725, A030726.

Cf. A030717.

nn = 30; m = 2; c[] = 0; k = a[1] = c[1] = 1; Reap[Do[w = Union@ Array[a, k]; Do[Set[a[j + k], c[w[[j]]]]; If[a[j + k] == m, Sow[m]], {j, Length[w]}]; Do[c[a[j + k]]++, {j, Length[w]}]; k += Length[w], {n, nn}] ][[-1, -1]] (* _Michael De Vlieger, Nov 19 2022 *)

A358466 Number of 1's that appeared by n-th step when constructing A030717.

Original entry on oeis.org

1, 2, 2, 3, 3, 4, 4, 5, 6, 7, 8, 8, 8, 10, 14, 17, 23, 30, 38, 49, 62, 77, 94, 110, 129, 149, 172, 195, 218, 241, 266, 293, 323, 356, 389, 424, 461, 500, 545, 593, 641, 688, 737, 787, 839, 896, 957, 1021, 1085, 1152, 1219, 1291, 1368, 1447, 1527, 1611, 1697, 1788, 1879, 1974, 2074, 2181, 2290, 2401, 2519

Offset: 1

Seiichi Manyama, Nov 18 2022

nonn

Michael De Vlieger, Table of n, a(n) for n = 1..1024

Partial sums of A358467.

Cf. A030717, A030723, A358468.

nn = 94; c[] = 0; k = a[1] = c[1] = 1; Accumulate@ Prepend[Reap[Do[w = Union@ Array[a, k]; t = 0; Do[Set[a[j + k], c[w[[j]]]]; If[a[j + k] == 1, t++], {j, Length[w]}]; Do[c[a[j + k]]++, {j, Length[w]}]; k += Length[w]; Sow[t], {n, nn}]][[-1, -1]], 1] (* _Michael De Vlieger, Nov 18 2022 *)

def A(k, n)
  a = []
  ary = [1]
  n.times{
    a << ary.count(k)
    ary += ary.uniq.sort.map{|i| ary.count(i)}
  }
  a
end
p A(1, 100)

A358467 Number of 1's that appeared in the n-th step when constructing A030717.

Original entry on oeis.org

1, 1, 0, 1, 0, 1, 0, 1, 1, 1, 1, 0, 0, 2, 4, 3, 6, 7, 8, 11, 13, 15, 17, 16, 19, 20, 23, 23, 23, 23, 25, 27, 30, 33, 33, 35, 37, 39, 45, 48, 48, 47, 49, 50, 52, 57, 61, 64, 64, 67, 67, 72, 77, 79, 80, 84, 86, 91, 91, 95, 100, 107, 109, 111, 118, 123, 127, 134, 138, 143, 148, 153, 158, 165, 171, 176, 179, 184, 187, 192, 195

Offset: 1

Seiichi Manyama, Nov 18 2022

nonn

Also first differences of A358466.

Michael De Vlieger, Table of n, a(n) for n = 1..1024

Cf. A030717, A358466, A358469.

nn = 94; c[] = 0; k = a[1] = c[1] = 1; Prepend[Reap[Do[w = Union@ Array[a, k]; t = 0; Do[Set[a[j + k], c[w[[j]]]]; If[a[j + k] == 1, t++], {j, Length[w]}]; Do[c[a[j + k]]++, {j, Length[w]}]; k += Length[w]; Sow[t], {n, nn}]][[-1, -1]], 1] (* _Michael De Vlieger, Nov 18 2022 *)

A358468 Number of 2's that appeared by n-th step when constructing A030717.

Original entry on oeis.org

0, 0, 1, 2, 3, 3, 3, 3, 4, 6, 8, 9, 10, 11, 12, 14, 16, 19, 23, 26, 27, 29, 32, 38, 43, 48, 51, 56, 63, 71, 79, 87, 94, 101, 109, 116, 125, 133, 140, 149, 161, 174, 188, 202, 217, 233, 250, 266, 284, 304, 326, 347, 369, 392, 418, 444, 471, 499, 530, 561, 593, 625, 658, 692, 726, 761, 797, 833, 869, 906, 944

Offset: 1

Seiichi Manyama, Nov 18 2022

nonn

Michael De Vlieger, Table of n, a(n) for n = 1..1024
Michael De Vlieger, Plot of the n-th row of A030717 n = 1..512, showing 2's in red and all other terms in blue.

Partial sums of A358469.

Cf. A030717, A358466.

nn = 94; c[] = 0; k = a[1] = c[1] = 1; Accumulate@ Prepend[Reap[Do[w = Union@ Array[a, k]; t = 0; Do[Set[a[j + k], c[w[[j]]]]; If[a[j + k] == 2, t++], {j, Length[w]}]; Do[c[a[j + k]]++, {j, Length[w]}]; k += Length[w]; Sow[t], {n, nn}]][[-1, -1]], 0] (* _Michael De Vlieger, Nov 18 2022 *)

def A(k, n)
  a = []
  ary = [1]
  n.times{
    a << ary.count(k)
    ary += ary.uniq.sort.map{|i| ary.count(i)}
  }
  a
end
p A(2, 100)

A358469 Number of 2's that appeared in the n-th step when constructing A030717.

Original entry on oeis.org

0, 0, 1, 1, 1, 0, 0, 0, 1, 2, 2, 1, 1, 1, 1, 2, 2, 3, 4, 3, 1, 2, 3, 6, 5, 5, 3, 5, 7, 8, 8, 8, 7, 7, 8, 7, 9, 8, 7, 9, 12, 13, 14, 14, 15, 16, 17, 16, 18, 20, 22, 21, 22, 23, 26, 26, 27, 28, 31, 31, 32, 32, 33, 34, 34, 35, 36, 36, 36, 37, 38, 37, 38, 38, 38, 39, 41, 42, 44, 45, 47, 47, 48, 49, 49, 49, 51, 53, 56, 56, 58, 60, 61, 62, 65

Offset: 1

Seiichi Manyama, Nov 18 2022

nonn

Also first differences of A358468.

Michael De Vlieger, Table of n, a(n) for n = 1..1024

Cf. A030717, A358467, A358468.

nn = 94; c[] = 0; k = a[1] = c[1] = 1; Prepend[Reap[Do[w = Union@ Array[a, k]; t = 0; Do[Set[a[j + k], c[w[[j]]]]; If[a[j + k] == 2, t++], {j, Length[w]}]; Do[c[a[j + k]]++, {j, Length[w]}]; k += Length[w]; Sow[t], {n, nn}]][[-1, -1]], 0] (* _Michael De Vlieger, Nov 18 2022 *)

A358423 Numbers k such that A030717(k) = 5.

Original entry on oeis.org

16, 18, 68, 76, 80, 89, 90, 93, 105, 109, 123, 143, 168, 286, 322, 366, 405, 448, 493, 494, 540, 541, 543, 591, 593, 646, 702, 770, 826, 832, 891, 893, 897, 960, 962, 966, 1032, 1034, 1039, 1110, 1112, 1117, 1193, 1195, 1200, 1279, 1281, 1286, 1367, 1369, 1374, 1459, 1461, 1466, 1554, 1556

Offset: 1

Seiichi Manyama, Nov 15 2022

nonn

Michael De Vlieger, Table of n, a(n) for n = 1..8044 (first 1000 term by Seiichi Manyama)
Michael De Vlieger, Plot a(s(j) + k - 1) at (j,k) for j = 1..512 and s = partial sums of A030719, showing a(m) = 5 in red, a(m) = 1 in medium blue for reference, and other values of a(m) in light blue

Cf. A030723, A030724, A030725, A030726, A358424, A358425.

Cf. A030717.

t = {{1}}; Do[AppendTo[t, BinCounts[#, {1, Max[#] + 1}] &[Flatten[t]]], {44}]; Position[DeleteCases[Flatten[t], 0], 5][[All, 1]] (* Michael De Vlieger, Nov 15 2022, after Peter J. C. Moses at A030717 *)

cp's OEIS Frontend

A030719 Number of new terms at stage n in the formation of A030717.

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A030723 Numbers k such that A030717(k) = 1.

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A030725 Numbers k such that A030717(k) = 3.

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A030726 Numbers k such that A030717(k) = 4.

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Extensions

A030724 Numbers k such that A030717(k) = 2.

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A358466 Number of 1's that appeared by n-th step when constructing A030717.

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A358467 Number of 1's that appeared in the n-th step when constructing A030717.

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A358468 Number of 2's that appeared by n-th step when constructing A030717.

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Mathematica

Ruby

A358469 Number of 2's that appeared in the n-th step when constructing A030717.

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Programs

Mathematica

A358423 Numbers k such that A030717(k) = 5.