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.

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A353841 Length of the trajectory of the partition run-sum transformation of n, using Heinz numbers; a(1) = 0.

Original entry on oeis.org

0, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 3, 1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 1, 2, 2, 1, 2, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 1, 2, 2, 1, 1, 2, 2, 2, 1, 2, 1, 2, 1, 2, 1, 1, 1, 3, 1, 1, 3, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 1, 2, 2, 1, 1, 4, 1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 1, 2, 1, 2, 2, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 3
Offset: 1

Views

Author

Gus Wiseman, May 25 2022

Keywords

Comments

The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.
Starting with n, this is one plus the number of times one must apply A353832 to reach a squarefree number.
Also Kimberling's depth statistic (defined in A237685 and A237750) plus one.

Examples

			The trajectory for a(1080) = 4 is the following, with prime indices shown on the right:
  1080: {1,1,1,2,2,2,3}
   325: {3,3,6}
   169: {6,6}
    37: {12}
The trajectory for a(87780) = 5 is the following, with prime indices shown on the right:
  87780: {1,1,2,3,4,5,8}
  65835: {2,2,3,4,5,8}
  51205: {3,4,4,5,8}
  19855: {3,5,8,8}
   2915: {3,5,16}
The trajectory for a(39960) = 5 is the following, with prime indices shown on the right:
  39960: {1,1,1,2,2,2,3,12}
  12025: {3,3,6,12}
   6253: {6,6,12}
   1369: {12,12}
     89: {24}

Links

Crossrefs

Positions of 1's are A005117.
The version for run-lengths instead of sums is A182850 or A323014.
Positions of first appearances are A353743.
These are the row-lengths of A353840.
Other sequences pertaining to this trajectory are A353842-A353845.
Counting partitions by this statistic gives A353846.
The version for compositions is A353854, run-lengths of A353853.
A001222 counts prime factors, distinct A001221.
A005811 counts runs in binary expansion.
A056239 adds up prime indices, row sums of A112798 and A296150.
A300273 ranks collapsible partitions, counted by A275870.
A318928 gives runs-resistance of binary expansion.
A353832 represents the operation of taking run-sums of a partition.
A353833 ranks partitions with all equal run-sums, counted by A304442.
A353835 counts distinct run-sums of prime indices, weak A353861.
A353838 ranks partitions with all distinct run-sums, counted by A353837.
A353866 ranks rucksack partitions, counted by A353864.
Cf. A008966, A071625, A073093, A181819, A182857, A325239, A325277, A325278, A353834, A353847, A353865, A353867.

Programs

  • Mathematica
    Table[If[n==1,0,Length[NestWhileList[Times@@Prime/@Cases[If[#==1,{},FactorInteger[#]],{p_,k_}:>PrimePi[p]*k]&,n,!SquareFreeQ[#]&]]],{n,100}]
  • PARI
    pis_to_runs(n) = { my(runs=List([]), f=factor(n)); for(i=1,#f~,while(f[i,2], listput(runs,primepi(f[i,1])); f[i,2]--)); (runs); };
A353832(n) = if(1==n,n,my(pruns = pis_to_runs(n), m=1, runsum=pruns[1]); for(i=2,#pruns,if(pruns[i] == pruns[i-1], runsum += pruns[i], m *= prime(runsum); runsum = pruns[i])); (m*prime(runsum)));
A353841(n) = if(1==n,0,for(i=1,oo,if(issquarefree(n), return(i), n = A353832(n)))); \\ Antti Karttunen, Jan 20 2025

Formula

a(1) = 0, and for n > 1, if A008966(n) = 1 [n is in A005117], a(n) = 1, otherwise a(n) = 1+a(A353832(n)). [See comments] - Antti Karttunen, Jan 20 2025

Extensions

More terms from Antti Karttunen, Jan 20 2025

A353931 Least run-sum of the prime indices of n.

Original entry on oeis.org

0, 1, 2, 2, 3, 1, 4, 3, 4, 1, 5, 2, 6, 1, 2, 4, 7, 1, 8, 2, 2, 1, 9, 2, 6, 1, 6, 2, 10, 1, 11, 5, 2, 1, 3, 2, 12, 1, 2, 3, 13, 1, 14, 2, 3, 1, 15, 2, 8, 1, 2, 2, 16, 1, 3, 3, 2, 1, 17, 2, 18, 1, 4, 6, 3, 1, 19, 2, 2, 1, 20, 3, 21, 1, 2, 2, 4, 1, 22, 3, 8, 1
Offset: 1

Views

Author

Gus Wiseman, Jun 07 2022

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.
Every sequence can be uniquely split into a sequence of non-overlapping runs. For example, the runs of (2,2,1,1,1,3,2,2) are ((2,2),(1,1,1),(3),(2,2)), with sums (4,3,3,4).

Examples

			The prime indices of 72 are {1,1,1,2,2}, with run-sums {3,4}, so a(72) = 3.

Crossrefs

Positions of first appearances are A008578.
For run-lengths instead of run-sums we have A051904, greatest A051903.
For run-sums and binary expansion we have A144790, greatest A038374.
For run-lengths and binary expansion we have A175597, greatest A043276.
Distinct run-sums are counted by A353835, weak A353861.
The greatest run-sum is given by A353862.
A001222 counts prime factors, distinct A001221.
A005811 counts runs in binary expansion.
A056239 adds up prime indices, row sums of A112798 and A296150.
A124010 gives prime signature, sorted A118914.
A304442 counts partitions with all equal run-sums, compositions A353851.
A353832 represents the operation of taking run-sums of a partition.
A353833 ranks partitions with all equal run sums, nonprime A353834.
A353838 ranks partitions with all distinct run-sums, counted by A353837.
A353840-A353846 pertain to partition run-sum trajectory.
Cf. A067340, A073093, A181819, A300273, A316413, A353839, A353866.

Programs

  • Mathematica
    Table[Min@@Cases[FactorInteger[n],{p_,k_}:>PrimePi[p]*k],{n,100}]

A382915 Number of integer partitions of n having no permutation with all equal run-lengths.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 2, 4, 4, 9, 11, 18, 21, 34, 41, 55, 69, 98, 120, 160, 189, 249, 309, 396, 472, 605, 734, 913, 1099, 1371, 1632, 2021, 2406, 2937, 3514, 4251, 5039, 6101, 7221, 8646, 10205, 12209, 14347, 17086, 20041, 23713, 27807, 32803, 38262, 45043, 52477, 61471, 71496
Offset: 0

Views

Author

Gus Wiseman, Apr 12 2025

Keywords

Examples

			The partition y = (2,2,1,1,1) has permutations and run-lengths:
  (2,2,1,1,1) (2,3)
  (2,1,2,1,1) (1,1,1,2)
  (2,1,1,2,1) (1,2,1,1)
  (2,1,1,1,2) (1,3,1)
  (1,2,2,1,1) (1,2,2)
  (1,2,1,2,1) (1,1,1,1,1)
  (1,2,1,1,2) (1,1,2,1)
  (1,1,2,2,1) (2,2,1)
  (1,1,2,1,2) (2,1,1,1)
  (1,1,1,2,2) (3,2)
Since (1,2,1,2,1) has all equal run-lengths (1,1,1,1,1), y is not counted under a(7).
The a(5) = 1 through a(10) = 11 partitions:
  (2111)  (3111)   (2221)    (5111)     (3222)      (3331)
          (21111)  (4111)    (41111)    (6111)      (4222)
                   (31111)   (311111)   (22221)     (7111)
                   (211111)  (2111111)  (51111)     (61111)
                                        (321111)    (421111)
                                        (411111)    (511111)
                                        (2211111)   (3211111)
                                        (3111111)   (4111111)
                                        (21111111)  (22111111)
                                                    (31111111)
                                                    (211111111)

Crossrefs

The complement for distinct run-lengths is A239455, ranked by A351294.
For distinct instead of equal run-lengths we have A351293, ranked by A351295.
These partitions are ranked by A382879, by signature A382914.
The complement is counted by A383013.
A000041 counts integer partitions, strict A000009.
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.
A382857 counts permutations of prime indices with equal run-lengths.
Cf. A003242, A047966, A238279, A329739, A351201, A351290, A351596, A382773.

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[n],Select[Permutations[#],SameQ@@Length/@Split[#]&]=={}&]],{n,0,15}]

Extensions

More terms from Bert Dobbelaere, Apr 26 2025

A383015 Numbers whose prime indices have more than one permutation with all equal run-sums.

Original entry on oeis.org

12, 40, 63, 112, 144, 325, 351, 352, 675, 832, 931, 1008, 1539, 1600, 1728, 2176, 2875, 3509, 3969, 4864, 6253, 7047, 7056, 8775, 9072, 11776, 12427, 12544, 12691, 16128, 19133, 20736, 20800, 22464, 23125, 26973, 29403, 29696, 32269, 43200, 49392, 57967, 59711
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.
All terms appear to have even sum of prime indices.

Examples

			The terms together with their prime indices begin:
     12: {1,1,2}
     40: {1,1,1,3}
     63: {2,2,4}
    112: {1,1,1,1,4}
    144: {1,1,1,1,2,2}
    325: {3,3,6}
    351: {2,2,2,6}
    352: {1,1,1,1,1,5}
    675: {2,2,2,3,3}
    832: {1,1,1,1,1,1,6}
    931: {4,4,8}
   1008: {1,1,1,1,2,2,4}
   1539: {2,2,2,2,8}
   1600: {1,1,1,1,1,1,3,3}
   1728: {1,1,1,1,1,1,2,2,2}

Crossrefs

Compositions of this type are counted by A353851, ranked by A353848.
Positions of terms > 1 in A382877, zeros A383100 (complement A383014).
For run-lengths instead of sums we have A383089, counted by A383090.
The complement for run-lengths instead of sums is A383091, counted by A383092
Partitions of this type are counted by A383097.
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, A329738, A353833, A354584, A381636, A381871, A382857, A382876, A382879.

Programs

  • Mathematica
    Select[Range[100],Length[Select[Permutations[PrimePi/@Join@@ConstantArray@@@FactorInteger[#]],SameQ@@Total/@Split[#]&]]>1&]

A383097 Number of integer partitions of n having more than one permutation with all equal run-sums.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 1, 0, 3, 0, 1, 0, 7, 0, 1, 0, 9, 0, 7, 0, 12, 0, 1, 0, 38, 0, 1, 1, 18, 0, 38, 0, 32, 0, 1, 0, 90, 0, 1, 0, 71, 0, 78, 0, 33, 10, 1, 0, 228, 0, 31, 0, 42, 0, 156, 0, 123, 0, 1, 0, 447, 0, 1, 16, 146, 0, 222, 0, 63, 0, 102, 0, 811, 0, 1, 29, 75, 0, 334, 0
Offset: 0

Views

Author

Gus Wiseman, Apr 17 2025

Keywords

Examples

			The a(27) = 1 partition is: (9,3,3,3,1,1,1,1,1,1,1,1,1).
The a(4) = 1 through a(16) = 9 partitions (empty columns not shown):
  (211)  (3111)  (422)     (511111)  (633)        (71111111)  (844)
                 (41111)             (6222)                   (82222)
                 (221111)            (33222)                  (442222)
                                     (4221111)                (44221111)
                                     (6111111)                (422221111)
                                     (33111111)               (811111111)
                                     (222111111)              (4411111111)
                                                              (42211111111)
                                                              (222211111111)

Crossrefs

These partitions are ranked by A383015, positions of terms > 1 in A382877.
For run-lengths instead of sums we have A383090, ranks A383089, unique A383094.
The complement is A383095 + A383096, ranks A383099 \/ A383100.
For any positive number of permutations we have A383098, ranks A383110.
Counting and ranking partitions by run-lengths and run-sums:
- constant: A047966 (ranks A072774), sums A304442 (ranks A353833)
- distinct: A098859 (ranks A130091), sums A353837 (ranks A353838)
- weakly decreasing: A100882 (ranks A242031), sums A304405 (ranks A357875)
- weakly increasing: A100883 (ranks A304678), sums A304406 (ranks A357861)
- strictly decreasing: A100881 (ranks A304686), sums A304428 (ranks A357862)
- strictly increasing: A100471 (ranks A334965), sums A304430 (ranks A357864)
A275870 counts collapsible partitions, ranks A300273.
A326534 ranks multiset partitions with a common sum, counted by A321455, normal A326518.
A353851 counts compositions with all equal run-sums, ranks A353848.
A382876 counts permutations of prime indices with distinct run-sums, zeros A381636.
Cf. A006171, A329738, A353832, A353839, A353850, A353932, A354584, A381717, A382076.

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[n],Length[Select[Permutations[#],SameQ@@Total/@Split[#]&]]>1&]],{n,0,15}]

Extensions

More terms from Bert Dobbelaere, Apr 26 2025

A383099 Numbers whose prime indices have exactly one permutation with all equal run-sums.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19, 23, 25, 27, 29, 31, 32, 36, 37, 41, 43, 47, 48, 49, 53, 59, 61, 64, 67, 71, 73, 79, 81, 83, 89, 97, 101, 103, 107, 109, 113, 121, 125, 127, 128, 131, 137, 139, 149, 151, 157, 163, 167, 169, 173, 179, 181, 191, 193
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 terms together with their prime indices begin:
    1: {}
    2: {1}
    3: {2}
    4: {1,1}
    5: {3}
    7: {4}
    8: {1,1,1}
    9: {2,2}
   11: {5}
   13: {6}
   16: {1,1,1,1}
   17: {7}
   19: {8}
   23: {9}
   25: {3,3}
   27: {2,2,2}
   29: {10}
   31: {11}
   32: {1,1,1,1,1}
   36: {1,1,2,2}
   37: {12}
   41: {13}

Crossrefs

For distinct instead of equal run-sums we have A000961, counted by A000005.
These are the positions of 1 in A382877.
For more than one choice we have A383015.
Partitions of this type are counted by A383095.
For no choices we have A383100, counted by A383096.
For at least one choice we have A383110, counted by A383098, see A383013.
For run-lengths instead of sums we have A383112 = positions of 1 in A382857.
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. A000720, A001221, A001222, A351294, A353832, A353838, A353932, A354584, A382876, A383091.

Programs

  • Mathematica
    Select[Range[100], Length[Select[Permutations[PrimePi/@Join @@ ConstantArray@@@FactorInteger[#]], SameQ@@Total/@Split[#]&]]==1&]

Formula

The complement is A383015 \/ A383100, for run-lengths A382879 \/ A383089.

A353862 Greatest run-sum of the prime indices of n.

Original entry on oeis.org

0, 1, 2, 2, 3, 2, 4, 3, 4, 3, 5, 2, 6, 4, 3, 4, 7, 4, 8, 3, 4, 5, 9, 3, 6, 6, 6, 4, 10, 3, 11, 5, 5, 7, 4, 4, 12, 8, 6, 3, 13, 4, 14, 5, 4, 9, 15, 4, 8, 6, 7, 6, 16, 6, 5, 4, 8, 10, 17, 3, 18, 11, 4, 6, 6, 5, 19, 7, 9, 4, 20, 4, 21, 12, 6, 8, 5, 6, 22, 4, 8
Offset: 1

Views

Author

Gus Wiseman, May 23 2022

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.
A run-sum of a sequence is the sum of any maximal consecutive constant subsequence.

Examples

			The prime indices of 72 are {1,1,1,2,2}, with run-sums {3,4}, so a(72) = 4.

Crossrefs

Positions of first appearances are A008578.
For binary expansion we have A038374, least A144790.
For run-lengths instead of run-sums we have A051903.
Distinct run-sums are counted by A353835, weak A353861.
The least run-sum is given by A353931.
A001222 counts prime factors, distinct A001221.
A005811 counts runs in binary expansion.
A056239 adds up prime indices, row sums of A112798 and A296150.
A124010 gives prime signature, sorted A118914.
A300273 ranks collapsible partitions, counted by A275870.
A304442 counts partitions with all equal run-sums, compositions A353851.
A353832 represents the operation of taking run-sums of a partition.
A353833 ranks partitions with all equal run sums, nonprime A353834.
A353838 ranks partitions with all distinct run-sums, counted by A353837.
A353840-A353846 pertain to partition run-sum trajectory.
Cf. A067340, A073093, A181819, A182857, A316413, A353836, A353839, A353848, A353852, A353866.

Programs

  • Mathematica
    Table[Max@@Cases[FactorInteger[n],{p_,k_}:>PrimePi[p]*k],{n,100}]

A383095 Number of integer partitions of n having exactly one permutation with all equal run-sums.

Original entry on oeis.org

1, 1, 2, 2, 3, 2, 6, 2, 4, 5, 6, 2, 12, 2, 6, 8, 5, 2, 20, 2, 12, 8, 6, 2, 20, 5, 6, 12, 12, 2, 34, 2, 6, 8, 6, 8, 45, 2, 6, 8, 20, 2, 34, 2, 12, 28, 6, 2, 30, 5, 20, 8, 12, 2, 52, 8, 20, 8, 6, 2, 78, 2, 6, 28, 7, 8, 34, 2, 12, 8, 34, 2, 80, 2, 6, 28, 12, 8, 34, 2, 30, 25
Offset: 0

Views

Author

Gus Wiseman, Apr 16 2025

Keywords

Examples

			The partition (2,2,1,1) has permutation (2,1,1,2) so is counted under a(6).
The a(1) = 1 through a(10) = 6 partitions (A=10):
  1  2   3    4     5      6       7        8         9          A
     11  111  22    11111  33      1111111  44        333        55
              1111         222              2222      33111      22222
                           2211             11111111  3111111    2221111
                           21111                      111111111  22111111
                           111111                                1111111111

Crossrefs

For distinct instead of equal run-sums we have A000005.
For run-lengths instead of sums we have A383094.
The complement is counted by A383096 + A383097, ranks A383100 \/ A383015.
These partitions are ranked by A383099 = positions of 1 in A382877.
Counting and ranking partitions by run-lengths and run-sums:
- constant: A047966 (ranks A072774), sums A304442 (ranks A353833)
- distinct: A098859 (ranks A130091), sums A353837 (ranks A353838)
- weakly decreasing: A100882 (ranks A242031), sums A304405 (ranks A357875)
- weakly increasing: A100883 (ranks A304678), sums A304406 (ranks A357861)
- strictly decreasing: A100881 (ranks A304686), sums A304428 (ranks A357862)
- strictly increasing: A100471 (ranks A334965), sums A304430 (ranks A357864)
A275870 counts collapsible partitions, ranks A300273.
A326534 ranks multiset partitions with a common sum, counted by A321455, normal A326518.
A353851 counts compositions with all equal run-sums, ranks A353848.
A383098 counts partitions with a permutation having all equal run-sums, ranks A383110.
Cf. A329738, A353839, A353832, A353932, A354584, A381636, A381871, A382076, A382876.

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[n], Length[Select[Permutations[#], SameQ@@Total/@Split[#]&]]==1&]],{n,0,15}]

Extensions

More terms from Bert Dobbelaere, Apr 26 2025

A383098 Number of integer partitions of n having at least one permutation with all equal run-sums.

Original entry on oeis.org

1, 1, 2, 2, 4, 2, 7, 2, 7, 5, 7, 2, 19, 2, 7, 8, 14, 2, 27, 2, 24, 8, 7, 2, 58, 5, 7, 13, 30, 2, 72, 2, 38, 8, 7, 8, 135, 2, 7, 8, 91, 2, 112, 2, 45, 38, 7, 2, 258, 5, 51, 8, 54, 2, 208, 8, 143, 8, 7, 2, 525, 2, 7, 44, 153, 8, 256, 2, 75, 8, 136, 2, 891, 2, 7, 57, 87, 8
Offset: 0

Views

Author

Gus Wiseman, Apr 17 2025

Keywords

Examples

			The partition (4,4,4,2,2,1,1,1,1) has permutations (4,2,2,4,1,1,1,1,4) and (4,1,1,1,1,4,2,2,4) so is counted under a(20).
The a(1) = 1 through a(10) = 7 partitions (A=10):
  1  2   3    4     5      6       7        8         9          A
     11  111  22    11111  33      1111111  44        333        55
              211          222              422       33111      22222
              1111         2211             2222      3111111    511111
                           3111             41111     111111111  2221111
                           21111            221111               22111111
                           111111           11111111             1111111111

Crossrefs

For distinct instead of equal run-sums we appear to have A382427.
For run-lengths instead of sums we have A383013, ranked by complement of A382879.
The case of a unique choice is A383095, ranks A383099 = positions of 1 in A382877.
The complement is counted by A383096, ranks A383100 = positions of 0 in A382877.
These partitions are ranked by A383110.
The case of more than one choice is A383097, ranks A383015.
Counting and ranking partitions by run-lengths and run-sums:
- constant: A047966 (ranks A072774), sums A304442 (ranks A353833)
- distinct: A098859 (ranks A130091), sums A353837 (ranks A353838)
- weakly decreasing: A100882 (ranks A242031), sums A304405 (ranks A357875)
- weakly increasing: A100883 (ranks A304678), sums A304406 (ranks A357861)
- strictly decreasing: A100881 (ranks A304686), sums A304428 (ranks A357862)
- strictly increasing: A100471 (ranks A334965), sums A304430 (ranks A357864)
A275870 counts collapsible partitions, ranks A300273.
A326534 ranks multiset partitions with a common sum, counted by A321455, normal A326518.
A353851 counts compositions with all equal run-sums, ranks A353848.
Cf. A006171, A329738, A353832, A353839, A353850, A353932, A354584, A382076, A382857, A382876, A383094, A383112.

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[n],Select[Permutations[#],SameQ@@Total/@Split[#]&]!={}&]],{n,0,15}]

Formula

a(n) = A383097(n) + A383095(n), ranks A383015 \/ A383099.

Extensions

More terms from Bert Dobbelaere, Apr 26 2025

A383110 Numbers whose prime indices have a permutation with all equal run-sums.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 11, 12, 13, 16, 17, 19, 23, 25, 27, 29, 31, 32, 36, 37, 40, 41, 43, 47, 48, 49, 53, 59, 61, 63, 64, 67, 71, 73, 79, 81, 83, 89, 97, 101, 103, 107, 109, 112, 113, 121, 125, 127, 128, 131, 137, 139, 144, 149, 151, 157, 163, 167, 169, 173
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 144 are {1,1,1,1,2,2}, with permutations with equal run sums (1,1,1,1,2,2), (1,1,2,1,1,2), (2,1,1,2,1,1), (2,2,1,1,1,1), so 144 is in the sequence.
The terms together with their prime indices begin:
   1: {}
   2: {1}
   3: {2}
   4: {1,1}
   5: {3}
   7: {4}
   8: {1,1,1}
   9: {2,2}
  11: {5}
  12: {1,1,2}
  13: {6}
  16: {1,1,1,1}
  17: {7}
  19: {8}
  23: {9}
  25: {3,3}
  27: {2,2,2}
  29: {10}
  31: {11}
  32: {1,1,1,1,1}
  36: {1,1,2,2}
  37: {12}

Crossrefs

For distinct run-sums we appear to have complement of A381636 (counted by A381717).
These are the positions of positive terms in A382877.
For run-lengths instead of sums we have complement of A382879, counted by A383013.
For more than one choice we have A383015.
Partitions of this type are counted by A383098.
For a unique choice we have A383099, counted by A383095.
The complement is A383100, counted by A383096.
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. A351295, A353832, A353837, A353838, A354584, A381871, A382857, A382876, A383097.

Programs

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

Formula

Equals A383015 \/ A383099, counted by A353851 \/ A383095.
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