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 1-4 of 4 results.

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}]

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).

A382773 Number of ways to permute a multiset whose multiplicities are the prime indices of n so that the run-lengths are all different.

Original entry on oeis.org

1, 1, 1, 0, 1, 2, 1, 0, 0, 2, 1, 0, 1, 2, 2, 0, 1, 0, 1, 0, 4, 4, 1, 0, 4, 4, 0, 0, 1, 6, 1, 0, 4, 6, 4, 0, 1, 6, 4, 0, 1, 6, 1, 0, 0, 8, 1, 0, 4, 0, 6, 0, 1, 0, 6, 0, 6, 8, 1, 0, 1, 10, 0, 0, 8, 6, 1, 0, 8, 6, 1, 0, 1, 10, 0, 0, 6, 6, 1, 0, 0, 12, 1, 0, 16
Offset: 1

Views

Author

Gus Wiseman, Apr 09 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}.

Examples

			The a(n) partitions for n = 6, 21, 30, 46:
  (1,1,2)  (1,1,1,1,2,2)  (1,1,1,2,2,3)  (1,1,1,1,1,1,1,1,1,2)
  (2,1,1)  (1,1,1,2,2,1)  (1,1,1,3,2,2)  (1,1,1,1,1,1,1,2,1,1)
           (1,2,2,1,1,1)  (2,2,1,1,1,3)  (1,1,1,1,1,1,2,1,1,1)
           (2,2,1,1,1,1)  (2,2,3,1,1,1)  (1,1,1,1,1,2,1,1,1,1)
                          (3,1,1,1,2,2)  (1,1,1,1,2,1,1,1,1,1)
                          (3,2,2,1,1,1)  (1,1,1,2,1,1,1,1,1,1)
                                         (1,1,2,1,1,1,1,1,1,1)
                                         (2,1,1,1,1,1,1,1,1,1)

Crossrefs

Positions of 1 are A008578.
For anti-run permutations we have A335125.
For just prime indices we have A382771, firsts A382772, equal A382857.
These permutations for factorials are counted by A382774, equal A335407.
For equal instead of distinct run-lengths we have A382858.
Positions of 0 are A382912, complement A382913.
A044813 lists numbers whose binary expansion has distinct run-lengths, equal A140690.
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, ranks A351294, conjugate A381432.
A329738 counts compositions with equal run-lengths, ranks A353744.
A329739 counts compositions with distinct run-lengths, ranks A351596.
A351293 counts non-Look-and-Say partitions, ranks A351295, conjugate A381433.
Cf. A000670, A000720, A003242, A048767, A130091, A181821, A238130, A305936, A351013, A351202, A382876, A382879.

Programs

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

Formula

a(n) = A382771(A181821(n)) = A382771(A304660(n)).

A382858 Number of ways to permute a multiset whose multiplicities are the prime indices of n so that the run-lengths are all equal.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 6, 4, 0, 1, 6, 1, 0, 1, 24, 1, 12, 1, 2, 1, 0, 1, 36, 4, 0, 36, 0, 1, 10, 1, 120, 0, 0, 1, 84, 1, 0, 0, 24, 1, 3, 1, 0, 38, 0, 1, 240, 6, 18, 0, 0, 1, 246, 0, 6, 0, 0, 1, 96, 1, 0, 30, 720, 1, 0, 1, 0, 0, 14, 1, 660, 1, 0, 74, 0, 1, 0, 1
Offset: 1

Views

Author

Gus Wiseman, Apr 09 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}.

Examples

			The a(9) = 4 permutations are:
  (1,1,2,2)
  (1,2,1,2)
  (2,1,2,1)
  (2,2,1,1)

Crossrefs

The anti-run case is A335125.
These permutations for factorials are counted by A335407, distinct A382774.
For distinct instead of equal run-lengths we have A382773.
For prime indices we have A382857 (firsts A382878), distinct A382771 (firsts A382772).
Positions of 0 are A382914, signature restriction of A382915.
A003963 gives product of prime indices.
A140690 lists numbers whose binary expansion has equal run-lengths, distinct A044813.
A047966 counts partitions with equal multiplicities, distinct A098859.
A056239 adds up prime indices, row sums of A112798.
A304442 counts partitions with equal run-sums, ranks A353833.
A329738 counts compositions with equal run-lengths, ranks A353744.
A329739 counts compositions with distinct run-lengths, ranks A351596, complement A351291.
A382913 ranks Look-and-Say partitions by signature, complement A382912.
Cf. A000720, A000961, A001221, A001222, A003242, A048767, A181821, A182854, A238130, A305936, A351202, A382879.

Programs

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

Formula

a(n) = A382857(A181821(n)) = A382857(A304660(n)).
Showing 1-4 of 4 results.

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