cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-10 of 22 results. Next

A381636 Numbers whose prime indices cannot be partitioned into constant blocks with distinct sums.

Original entry on oeis.org

12, 60, 63, 84, 120, 126, 132, 156, 204, 228, 252, 276, 300, 315, 325, 348, 372, 420, 444, 492, 504, 516, 560, 564, 588, 630, 636, 650, 660, 693, 708, 720, 732, 780, 804, 819, 840, 852, 876, 924, 931, 948, 975, 996, 1008, 1020, 1068, 1071, 1092, 1140, 1164
Offset: 1

Views

Author

Gus Wiseman, Mar 10 2025

Keywords

Comments

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.
Also numbers that cannot be written as a product of prime powers > 1 with distinct sums of prime indices (A056239).
Contains no squarefree numbers.
Conjecture: These are the zeros of A382876.

Examples

			The prime indices of 300 are {1,1,2,3,3}, with partitions into constant blocks:
  {{2},{1,1},{3,3}}
  {{1},{1},{2},{3,3}}
  {{2},{3},{3},{1,1}}
  {{1},{1},{2},{3},{3}}
but none of these has distinct block-sums, so 300 is in the sequence.
The terms together with their prime indices begin:
   12: {1,1,2}
   60: {1,1,2,3}
   63: {2,2,4}
   84: {1,1,2,4}
  120: {1,1,1,2,3}
  126: {1,2,2,4}
  132: {1,1,2,5}
  156: {1,1,2,6}
  204: {1,1,2,7}
  228: {1,1,2,8}
  252: {1,1,2,2,4}
  276: {1,1,2,9}
  300: {1,1,2,3,3}

Crossrefs

More on multiset partitions into constant blocks: A006171, A279784, A295935.
These are the positions of 0 in A381635, after taking block-sums A381716.
Partitions of this type are counted by A381717.
For strict instead of constant blocks we have A381806, zeros of A381633.
For equal instead of distinct block-sums we have A381871.
A000688 counts multiset partitions into constant, see A381455 (upper), A381453 (lower).
A001055 counts multiset partitions, see A317141 (upper), A300383 (lower).
A050361 counts multiset partitions into distinct constant blocks, after sums A381715.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798.
Cf. A000720, A001222, A005117, A050320, A059404, A213242, A293243, A299202, A300385, A381078, A381454, A381634.

Programs

  • Mathematica
    hwt[n_]:=Total[Cases[FactorInteger[n],{p_,k_}:>PrimePi[p]*k]];
pfacs[n_]:=If[n<=1,{{}},Join@@Table[(Prepend[#,d]&)/@Select[pfacs[n/d],Min@@#>=d&],{d,Select[Rest[Divisors[n]],PrimePowerQ]}]];
Select[Range[100],Select[pfacs[#],UnsameQ@@hwt/@#&]=={}&]

A383706 Number of ways to choose disjoint strict integer partitions, one of each prime index of n.

Original entry on oeis.org

1, 1, 1, 0, 2, 1, 2, 0, 0, 1, 3, 0, 4, 1, 1, 0, 5, 0, 6, 0, 2, 2, 8, 0, 2, 2, 0, 0, 10, 1, 12, 0, 2, 3, 2, 0, 15, 3, 2, 0, 18, 1, 22, 0, 0, 5, 27, 0, 2, 0, 3, 0, 32, 0, 3, 0, 4, 5, 38, 0, 46, 7, 0, 0, 4, 1, 54, 0, 5, 1, 64, 0, 76, 8, 0, 0, 3, 1, 89, 0, 0, 10
Offset: 1

Views

Author

Gus Wiseman, May 15 2025

Keywords

Comments

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

Examples

			The prime indices of 25 are (3,3), for which we have choices ((3),(2,1)) and ((2,1),(3)), so a(25) = 2.
The prime indices of 91 are (4,6), for which we have choices ((4),(6)), ((4),(5,1)), ((4),(3,2,1)), ((3,1),(6)), ((3,1),(4,2)), so a(91) = 5.
The prime indices of 273 are (2,4,6), for which we have choices ((2),(4),(6)), ((2),(4),(5,1)), ((2),(3,1),(6)), so a(273) = 3.

Crossrefs

Adding up over all integer partitions gives A279790, strict A279375.
Without disjointness we have A357982, non-strict version A299200.
For multiplicities instead of indices we have A382525.
Positions of 0 appear to be A382912, counted by A383710, odd case A383711.
Positions of positive terms are A382913, counted by A383708, odd case A383533.
Positions of 1 are A383707, counted by A179009.
The conjugate version is A384005.
A000041 counts integer partitions, strict A000009.
A048767 is the Look-and-Say transform, fixed points A048768, counted by A217605.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798.
A239455 counts Look-and-Say or section-sum partitions, ranks A351294 or A381432.
A351293 counts non-Look-and-Say or non-section-sum partitions, ranks A351295 or A381433.
Cf. A098859, A122111, A130091, A209229, A317141, A381454, A382771, A382876.

Programs

  • Mathematica
    pof[y_]:=Select[Join@@@Tuples[IntegerPartitions/@y], UnsameQ@@#&];
prix[n_]:=If[n==1,{}, Flatten[Cases[FactorInteger[n], {p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Length[pof[prix[n]]],{n,100}]

A382857 Number of ways to permute the prime indices of n so that the run-lengths are all equal.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 2, 2, 1, 0, 1, 2, 1, 1, 1, 6, 1, 1, 2, 2, 2, 4, 1, 2, 2, 0, 1, 6, 1, 1, 1, 2, 1, 0, 1, 1, 2, 1, 1, 0, 2, 0, 2, 2, 1, 6, 1, 2, 1, 1, 2, 6, 1, 1, 2, 6, 1, 1, 1, 2, 1, 1, 2, 6, 1, 0, 1, 2, 1, 6, 2, 2
Offset: 0

Views

Author

Gus Wiseman, Apr 09 2025

Keywords

Comments

The first x with a(x) > 1 but A382771(x) > 0 is a(216) = 4, A382771(216) = 4.
A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798, sum A056239.

Examples

			The prime indices of 216 are {1,1,1,2,2,2} and we have permutations:
  (1,1,1,2,2,2)
  (1,2,1,2,1,2)
  (2,1,2,1,2,1)
  (2,2,2,1,1,1)
so a(216) = 4.
The prime indices of 25920 are {1,1,1,1,1,1,2,2,2,2,3} and we have permutations:
  (1,2,1,2,1,2,1,2,1,3,1)
  (1,2,1,2,1,2,1,3,1,2,1)
  (1,2,1,2,1,3,1,2,1,2,1)
  (1,2,1,3,1,2,1,2,1,2,1)
  (1,3,1,2,1,2,1,2,1,2,1)
so a(25920) = 5.

Crossrefs

The restriction to signature representatives (A181821) is A382858, distinct A382773.
The restriction to factorials is A335407, distinct A382774.
For distinct instead of equal run-lengths we have A382771.
For run-sums instead of run-lengths we have A382877, distinct A382876.
Positions of first appearances are A382878.
Positions of 0 are A382879.
Positions of terms > 1 are A383089.
Positions of 1 are A383112.
A003963 gives product of prime indices.
A005811 counts runs in binary expansion.
A044813 lists numbers whose binary expansion has distinct run-lengths.
A056239 adds up prime indices, row sums of A112798.
A239455 counts Look-and-Say partitions, ranks A351294.
A304442 counts partitions with equal run-sums, ranks A353833.
A164707 lists numbers whose binary expansion has all equal run-lengths, distinct A328592.
A353744 ranks compositions with equal run-lengths, counted by A329738.
Cf. A000720, A000961, A001221, A001222, A003242, A008480, A047966, A238130, A238279, A351201, A351293, A351295.

Programs

  • Mathematica
    Table[Length[Select[Permutations[Join@@ConstantArray@@@FactorInteger[n]], SameQ@@Length/@Split[#]&]],{n,0,100}]

A381717 Number of integer partitions of n that cannot be partitioned into constant multisets with distinct block-sums.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 0, 1, 3, 2, 3, 6, 7, 10, 15, 15, 28, 37, 47, 64, 71, 97, 139, 173, 215, 273, 361, 439, 551, 691, 853, 1078, 1325, 1623, 2046, 2458, 2998, 3697, 4527, 5472, 6590, 7988, 9590, 11598, 13933, 16560, 19976, 23822, 28420, 33797, 40088, 47476, 56369, 66678
Offset: 0

Views

Author

Gus Wiseman, Mar 16 2025

Keywords

Comments

Conjecture: Also the number of integer partitions of n having no permutation with all distinct run-sums, ranked by zeros of A382876. In other words, a partition has a permutation with all distinct run-sums iff it has a multiset partition into constant blocks with all distinct block-sums, where the run-sums of a sequence are obtained by splitting it into maximal runs and taking their sums.

Examples

			For y = (3,2,2,1) we have the multiset partition {{3},{2,2},{1}}, so y is not counted under a(8).
For y = (3,2,1,1,1) there are 3 multiset partitions into constant multisets:
  {{3},{2},{1,1,1}}
  {{3},{2},{1,1},{1}}
  {{3},{2},{1},{1},{1}}
but none of these has distinct block-sums, so y is counted under a(8).
For y = (3,3,1,1,1,1,1,1) we have multiset partitions:
  {{1},{3,3},{1,1,1,1,1}}
  {{1,1},{3,3},{1,1,1,1}}
  {{1},{1,1},{3,3},{1,1,1}}
so y is not counted under a(12).
The a(4) = 1 through a(13) = 10 partitions:
  211  .  .  3211  422    4221  6211   4322     633      5422
                   4211   5211  33211  7211     8211     6331
                   32111        42211  43211    43221    9211
                                       422111   44211    54211
                                       431111   53211    63211
                                       3221111  432111   333211
                                                4221111  432211
                                                         532111
                                                         4321111
                                                         42211111

Crossrefs

Twice-partitions of this type (constant with distinct) are counted by A279786.
Multiset partitions of this type are ranked by A326535 /\ A355743.
These partitions are ranked by A381636, zeros of A381635.
For strict instead of constant blocks we have A381990, see A381806, A381633, A382079.
For equal instead of distinct block-sums we have A381993.
A000041 counts integer partitions, strict A000009.
A000688 counts factorizations into prime powers, see A381455, A381453.
A001055 counts factorizations, strict A045778, see A317141, A300383.
A050361 counts factorizations into distinct prime powers.
Cf. A002846, A047966, A130091, A213242, A213427, A265947, A317142, A353864, A381716, A381991, A381992, A382876.

Programs

  • Mathematica
    mce[y_]:=Table[ConstantArray[y[[1]],#]&/@ptn,{ptn,IntegerPartitions[Length[y]]}];
Table[Length[Select[IntegerPartitions[n],Select[Join@@@Tuples[mce/@Split[#]],UnsameQ@@Total/@#&]=={}&]],{n,0,30}]

Extensions

a(37)-a(53) from Robert Price, Mar 31 2025

A382879 Positions of 0 in A382857 (permutations of prime indices with equal run-lengths).

Original entry on oeis.org

24, 40, 48, 54, 56, 80, 88, 96, 104, 112, 135, 136, 152, 160, 162, 176, 184, 189, 192, 208, 224, 232, 240, 248, 250, 272, 288, 296, 297, 304, 320, 328, 336, 344, 351, 352, 368, 375, 376, 384, 405, 416, 424, 448, 459, 464, 472, 480, 486, 488, 496, 513, 528, 536
Offset: 1

Views

Author

Gus Wiseman, Apr 09 2025

Keywords

Comments

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798, sum A056239.

Examples

			The terms together with their prime indices begin:
   24: {1,1,1,2}
   40: {1,1,1,3}
   48: {1,1,1,1,2}
   54: {1,2,2,2}
   56: {1,1,1,4}
   80: {1,1,1,1,3}
   88: {1,1,1,5}
   96: {1,1,1,1,1,2}
  104: {1,1,1,6}
  112: {1,1,1,1,4}
  135: {2,2,2,3}
  136: {1,1,1,7}
  152: {1,1,1,8}
  160: {1,1,1,1,1,3}

Crossrefs

For distinct instead of equal the complement is A351294, counted by A239455.
For distinct instead of equal we have A351295, counted by A351293.
For run-sums instead of run-lengths we have A383100, zeros of A382877, distinct A382876.
Positions of 0 in A382857 (firsts A382878), by signature A382858 (distinct A382773).
For prime signature instead of prime indices we have A382914.
Partitions of this type are counted by A382915.
The complement is counted by A383013.
A005811 counts runs in binary expansion.
A056239 adds up prime indices, row sums of A112798.
A297770 counts distinct runs in binary expansion.
A164707 lists numbers whose binary form has equal runs of ones, distinct A328592.
A304442 counts partitions with equal run-sums, ranks A353833.
A329739 counts compositions with distinct run-lengths, ranks A351290.
A353744 ranks compositions with equal run-lengths, distinct A351596 (complement A351291).
Cf. A000720, A000961, A001221, A001222, A003242, A008480, A047966, A130091, A238279, A351201.

Programs

  • Mathematica
    Select[Range[100], Select[Permutations[Join@@ConstantArray@@@FactorInteger[#]], SameQ@@Length/@Split[#]&]=={}&]

A382912 Numbers k such that row k of A305936 (a multiset whose multiplicities are the prime indices of k) has no permutation with all distinct run-lengths.

Original entry on oeis.org

4, 8, 9, 12, 16, 18, 20, 24, 27, 28, 32, 36, 40, 44, 45, 48, 50, 52, 54, 56, 60, 63, 64, 68, 72, 75, 76, 80, 81, 84, 88, 90, 92, 96, 98, 99, 100, 104, 108, 112, 116, 117, 120, 124, 125, 126, 128, 132, 135, 136, 140, 144, 148, 150, 152, 153, 156, 160, 162, 164
Offset: 1

Views

Author

Gus Wiseman, Apr 12 2025

Keywords

Comments

This described multiset (row n of A305936, Heinz number A181821) is generally not the same as the multiset of prime indices of n (A112798). For example, the prime indices of 12 are {1,1,2}, while a multiset whose multiplicities are {1,1,2} is {1,1,2,3}.
A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798, sum A056239.

Examples

			The terms, prime indices, and corresponding multisets begin:
   4:       {1,1} {1,2}
   8:     {1,1,1} {1,2,3}
   9:       {2,2} {1,1,2,2}
  12:     {1,1,2} {1,1,2,3}
  16:   {1,1,1,1} {1,2,3,4}
  18:     {1,2,2} {1,1,2,2,3}
  20:     {1,1,3} {1,1,1,2,3}
  24:   {1,1,1,2} {1,1,2,3,4}
  27:     {2,2,2} {1,1,2,2,3,3}
  28:     {1,1,4} {1,1,1,1,2,3}
  32: {1,1,1,1,1} {1,2,3,4,5}
  36:   {1,1,2,2} {1,1,2,2,3,4}
  40:   {1,1,1,3} {1,1,1,2,3,4}
  44:     {1,1,5} {1,1,1,1,1,2,3}
  45:     {2,2,3} {1,1,1,2,2,3,3}
  48: {1,1,1,1,2} {1,1,2,3,4,5}
  50:     {1,3,3} {1,1,1,2,2,2,3}
  52:     {1,1,6} {1,1,1,1,1,1,2,3}

Crossrefs

The Look-and-Say partition is ranked by A048767, listed by A381440.
Look-and-Say partitions are counted by A239455, ranks A351294.
Non-Look-and-Say partitions are counted by A351293.
For prime indices instead of signature we have A351295, conjugate A381433.
The complement is A382913.
For equal instead of distinct run-lengths we have A382914, see A382858, A382879, A382915.
A056239 adds up prime indices, row sums of A112798.
A329739 counts compositions with distinct run-lengths, ranks A351596, complement A351291.
A381431 lists the section-sum partition of n, ranks A381436, union A381432.
Cf. A000720, A001221, A001222, A181821, A305936, A335125, A351202, A382525, A382771, A382773, A382775, A382857.
Cf. A381636, A381717, A381871, A382876.

Programs

  • Mathematica
    nrmptn[n_]:=Join@@MapIndexed[Table[#2[[1]],{#1}]&,If[n==1,{}, Flatten[Cases[FactorInteger[n]//Reverse,{p_,k_} :> Table[PrimePi[p],{k}]]]]];
lasQ[y_]:=Select[Permutations[y], UnsameQ@@Length/@Split[#]&]!={};
Select[Range[100],Not@*lasQ@*nrmptn]

A382913 Numbers k such that row k of A305936 (a multiset whose multiplicities are the prime indices of k) has a permutation with all distinct run-lengths.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 25, 26, 29, 30, 31, 33, 34, 35, 37, 38, 39, 41, 42, 43, 46, 47, 49, 51, 53, 55, 57, 58, 59, 61, 62, 65, 66, 67, 69, 70, 71, 73, 74, 77, 78, 79, 82, 83, 85, 86, 87, 89, 91, 93, 94, 95, 97, 101, 102, 103
Offset: 1

Views

Author

Gus Wiseman, Apr 12 2025

Keywords

Comments

This described multiset (row n of A305936, Heinz number A181821) is generally not the same as the multiset of prime indices of n (A112798). For example, the prime indices of 12 are {1,1,2}, while a multiset whose multiplicities are {1,1,2} is {1,1,2,3}.
A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798, sum A056239.

Examples

			The terms, prime indices, and corresponding multisets begin:
   1:    {} {}
   2:   {1} {1}
   3:   {2} {1,1}
   5:   {3} {1,1,1}
   6: {1,2} {1,1,2}
   7:   {4} {1,1,1,1}
  10: {1,3} {1,1,1,2}
  11:   {5} {1,1,1,1,1}
  13:   {6} {1,1,1,1,1,1}
  14: {1,4} {1,1,1,1,2}
  15: {2,3} {1,1,1,2,2}
  17:   {7} {1,1,1,1,1,1,1}
  19:   {8} {1,1,1,1,1,1,1,1}
  21: {2,4} {1,1,1,1,2,2}
  22: {1,5} {1,1,1,1,1,2}
  23:   {9} {1,1,1,1,1,1,1,1,1}
  25: {3,3} {1,1,1,2,2,2}
  26: {1,6} {1,1,1,1,1,1,2}

Crossrefs

Look-and-Say partitions are counted by A239455, ranks A351294.
Non-Look-and-Say partitions are counted by A351293, ranks A351295.
For prime indices instead of signature we have A351294, conjugate A381432.
The Look-and-Say partition of n is listed by A381440, rank A048767.
The complement is A382912.
For equal run-lengths we have the complement of A382914, see A382858, A382879, A382915.
A044813 lists numbers whose binary expansion has distinct run-lengths.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798.
A329739 counts compositions with distinct run-lengths, ranks A351596.
A381431 ranks section-sum partition, listed by A381436.
Cf. A000720, A001221, A001222, A140690, A181821, A305936, A335125, A382525, A382771, A382773, A382775.
Cf. A381636, A381717, A381871, A382876.

Programs

  • Mathematica
    nrmptn[n_]:=Join@@MapIndexed[Table[#2[[1]],{#1}]&, If[n==1,{},Flatten[Cases[FactorInteger[n]//Reverse,{p_,k_} :> Table[PrimePi[p],{k}]]]]];
lasQ[y_]:=Select[Permutations[y], UnsameQ@@Length/@Split[#]&]!={};
Select[Range[100],lasQ@*nrmptn]

A382771 Number of ways to permute the prime indices of n so that the run-lengths are all different.

Original entry on oeis.org

1, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 2, 1, 0, 0, 1, 1, 2, 1, 2, 0, 0, 1, 2, 1, 0, 1, 2, 1, 0, 1, 1, 0, 0, 0, 0, 1, 0, 0, 2, 1, 0, 1, 2, 2, 0, 1, 2, 1, 2, 0, 2, 1, 2, 0, 2, 0, 0, 1, 0, 1, 0, 2, 1, 0, 0, 1, 2, 0, 0, 1, 2, 1, 0, 2, 2, 0, 0, 1, 2, 1, 0, 1, 0, 0, 0, 0
Offset: 1

Views

Author

Gus Wiseman, Apr 07 2025

Keywords

Comments

The first x with a(x) > 0 but A382857(x) > 1 is a(216) = 4, A382857(216) = 4.
A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798, sum A056239.

Examples

			The a(96) = 4 permutations are:
  (1,1,1,1,1,2)
  (1,1,1,2,1,1)
  (1,1,2,1,1,1)
  (2,1,1,1,1,1)
The a(216) = 4 permutations are:
  (1,1,2,2,2,1)
  (1,2,2,2,1,1)
  (2,1,1,1,2,2)
  (2,2,1,1,1,2)
The a(360) = 6 permutations are:
  (1,1,1,2,2,3)
  (1,1,1,3,2,2)
  (2,2,1,1,1,3)
  (2,2,3,1,1,1)
  (3,1,1,1,2,2)
  (3,2,2,1,1,1)

Crossrefs

Positions of 1 are A000961.
Positions of positive terms are A351294, conjugate A381432.
Positions of 0 are A351295, conjugate A381433, equal A382879.
Sorted positions of first appearances are A382772, equal A382878.
For prescribed signature we have A382773, equal A382858.
The restriction to factorials is A382774, equal A335407.
For equal instead of distinct run-lengths we have A382857.
For run-sums instead of run-lengths we have A382876, equal A382877.
Positions of terms > 1 are A383113.
A044813 lists numbers whose binary expansion has distinct run-lengths.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798.
A098859 counts partitions with distinct multiplicities, ordered A242882.
A239455 counts Look-and-Say partitions, complement A351293.
A329738 counts compositions with equal run-lengths, ranks A353744.
A329739 counts compositions with distinct run-lengths, ranks A351596.
Cf. A000720, A001221, A001222, A003242, A048767, A051903, A051904, A130091, A238130, A351013, A351202.

Programs

  • Mathematica
    Table[Length[Select[Permutations[Join@@ConstantArray@@@FactorInteger[n]],UnsameQ@@Length/@Split[#]&]],{n,30}]

Formula

a(A181821(n)) = a(A304660(n)) = A382773(n).
a(n!) = A382774(n).

A382877 Number of ways to permute the prime indices of n so that the run-sums are all equal.

Original entry on oeis.org

1, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 2, 1, 0, 0, 1, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 0, 0, 0, 1, 1, 0, 0, 2, 1, 0, 1, 0, 0, 0, 1, 1, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 2, 1, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 0
Offset: 1

Views

Author

Gus Wiseman, Apr 14 2025

Keywords

Comments

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798, sum A056239.

Examples

			The a(144) = 4 permutations of {1,1,1,1,2,2} are:
  (1,1,1,1,2,2)
  (1,1,2,1,1,2)
  (2,1,1,2,1,1)
  (2,2,1,1,1,1)
The a(1728) = 4 permutations are:
  (1,1,1,1,1,1,2,2,2)
  (1,1,2,1,1,2,1,1,2)
  (2,1,1,2,1,1,2,1,1)
  (2,2,2,1,1,1,1,1,1)

Crossrefs

Compositions of this type are counted by A353851, ranked by A353848.
For run-lengths instead of sums we have A382857 (zeros A382879), distinct A382771.
For distinct instead of equal run-sums we have A382876, counted by A353850.
Positions of terms > 1 are A383015.
Positions of 1 are A383099.
Positions of 0 are A383100 (complement A383110), counted by A383098.
A044813 lists numbers whose binary expansion has distinct run-lengths.
A056239 adds up prime indices, row sums of A112798.
A304442 counts compositions with equal run-sums, complement A382076.
A329739 counts compositions with distinct run-lengths, ranks A351596.
A353837 counts partitions with distinct run-sums, ranks A353838.
A353847 gives composition run-sum transformation, for partitions A353832.
A353932 lists run-sums of standard compositions.
Cf. A000720, A000961, A001221, A001222, A130091, A329738, A353833, A353852, A354584, A381871.

Programs

  • Mathematica
    Table[Length[Select[Permutations[PrimePi/@Join @@ ConstantArray@@@FactorInteger[n]], SameQ@@Total/@Split[#]&]],{n,100}]

A383100 Numbers whose prime indices have no permutation with all equal run-sums.

Original entry on oeis.org

6, 10, 14, 15, 18, 20, 21, 22, 24, 26, 28, 30, 33, 34, 35, 38, 39, 42, 44, 45, 46, 50, 51, 52, 54, 55, 56, 57, 58, 60, 62, 65, 66, 68, 69, 70, 72, 74, 75, 76, 77, 78, 80, 82, 84, 85, 86, 87, 88, 90, 91, 92, 93, 94, 95, 96, 98, 99, 100, 102, 104, 105, 106, 108
Offset: 1

Views

Author

Gus Wiseman, Apr 20 2025

Keywords

Comments

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798, sum A056239.

Examples

			The prime indices of 18 are {1,2,2}, with permutations (1,2,2), (2,1,2), (2,2,1), with run sums (1,4), (2,1,2), (4,1) respectively, so 18 is in the sequence.
The terms together with their prime indices begin:
    6: {1,2}
   10: {1,3}
   14: {1,4}
   15: {2,3}
   18: {1,2,2}
   20: {1,1,3}
   21: {2,4}
   22: {1,5}
   24: {1,1,1,2}
   26: {1,6}
   28: {1,1,4}
   30: {1,2,3}
   33: {2,5}
   34: {1,7}
   35: {3,4}
   38: {1,8}
   39: {2,6}
   42: {1,2,4}
   44: {1,1,5}
   45: {2,2,3}
   46: {1,9}
   50: {1,3,3}

Crossrefs

For distinct instead of equal run-sums we appear to have A381636, counted by A381717.
For run-lengths instead of sums we have A382879, counted by complement of A383013.
These are the positions of 0 in A382877.
For more than one choice we have A383015.
The complement is A383110, counted by A383098.
Partitions of this type are counted by A383096.
For a unique choice we have A383099, counted by A383095.
A056239 adds up prime indices, row sums of A112798.
A304442 counts partitions with equal run-sums, ranks A353833.
A353851 counts compositions with equal run-sums, ranks A353848.
Cf. A047966, A072774, A130091, A242031, A304686, A304678, A334965.
Cf. A351294, A351295, A353832, A353837, A353838, A354584, A381871, A382857, A382876, A383094, A383097.

Programs

  • Mathematica
    Select[Range[100], Length[Select[Permutations[PrimePi/@Join @@ ConstantArray@@@FactorInteger[#]], SameQ@@Total/@Split[#]&]]==0&]
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