A335838 -id:A335838 - cp's OEIS frontend

A335126 A multiset whose multiplicities are the prime indices of n is inseparable.

3, 5, 7, 10, 11, 13, 14, 17, 19, 21, 22, 23, 26, 28, 29, 31, 33, 34, 37, 38, 39, 41, 43, 44, 46, 47, 51, 52, 53, 55, 57, 58, 59, 61, 62, 65, 66, 67, 68, 69, 71, 73, 74, 76, 78, 79, 82, 83, 85, 86, 87, 88, 89, 91, 92, 93, 94, 95, 97, 101, 102, 103, 104, 106

Offset: 1

Gus Wiseman, Jul 01 2020

nonn

A multiset is separable if it has a permutation that is an anti-run, meaning there are no adjacent equal parts.

A multiset whose multiplicities are the prime indices of n (such as row n of A305936) is not the same as the multiset of prime indices of n. 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}.

			The sequence of terms together with the corresponding multisets begins:
   3: {1,1}
   5: {1,1,1}
   7: {1,1,1,1}
  10: {1,1,1,2}
  11: {1,1,1,1,1}
  13: {1,1,1,1,1,1}
  14: {1,1,1,1,2}
  17: {1,1,1,1,1,1,1}
  19: {1,1,1,1,1,1,1,1}
  21: {1,1,1,1,2,2}
  22: {1,1,1,1,1,2}
  23: {1,1,1,1,1,1,1,1,1}
  26: {1,1,1,1,1,1,2}
  28: {1,1,1,1,2,3}
  29: {1,1,1,1,1,1,1,1,1,1}

The complement is A335127.
Anti-run compositions are A003242.
Anti-runs are ranked by A333489.
Separable partitions are A325534.
Inseparable partitions are A325535.
Separable factorizations are A335434.
Inseparable factorizations are A333487.
Separable partitions are ranked by A335433.
Inseparable partitions are ranked by A335448.
Anti-run permutations of prime indices are A335452.
Patterns contiguously matched by compositions are A335457.
Cf. A025487, A056239, A106351, A112798, A114938, A181819, A181821, A278990, A292884, A335407, A335489, A335516, A335838.

nrmptn[n_]:=Join@@MapIndexed[Table[#2[[1]],{#1}]&,If[n==1,{},Flatten[Cases[FactorInteger[n]//Reverse,{p_,k_}:>Table[PrimePi[p],{k}]]]]];
Select[Range[100],Select[Permutations[nrmptn[#]],!MatchQ[#,{_,x_,x_,_}]&]=={}&]

A335434 Number of separable factorizations of n into factors > 1.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 4, 1, 2, 2, 3, 1, 4, 1, 4, 2, 2, 1, 6, 1, 2, 2, 4, 1, 5, 1, 5, 2, 2, 2, 8, 1, 2, 2, 6, 1, 5, 1, 4, 4, 2, 1, 10, 1, 4, 2, 4, 1, 6, 2, 6, 2, 2, 1, 11, 1, 2, 4, 6, 2, 5, 1, 4, 2, 5, 1, 15, 1, 2, 4, 4, 2, 5, 1, 10, 3, 2, 1, 11, 2

Offset: 1

Gus Wiseman, Jul 03 2020

nonn

A multiset is separable if it has a permutation that is an anti-run, meaning there are no adjacent equal parts.

			The a(n) factorizations for n = 2, 6, 16, 12, 30, 24, 36, 48, 60:
  2  6    16     12     30     24     36       48       60
     2*3  2*8    2*6    5*6    3*8    4*9      6*8      2*30
          2*2*4  3*4    2*15   4*6    2*18     2*24     3*20
                 2*2*3  3*10   2*12   3*12     3*16     4*15
                        2*3*5  2*2*6  2*2*9    4*12     5*12
                               2*3*4  2*3*6    2*3*8    6*10
                                      3*3*4    2*4*6    2*5*6
                                      2*2*3*3  3*4*4    3*4*5
                                               2*2*12   2*2*15
                                               2*2*3*4  2*3*10
                                                        2*2*3*5

The version for partitions is A325534.
The inseparable version is A333487.
The version for multisets with prescribed multiplicities is A335127.
Factorizations are A001055.
Anti-run compositions are A003242.
Inseparable partitions are A325535.
Anti-runs are ranked by A333489.
Separable partitions are ranked by A335433.
Inseparable partitions are ranked by A335448.
Anti-run permutations of prime indices are A335452.
Cf. A106351, A292884, A295370, A333628, A333755, A335463, A335125, A335126, A335407, A335457, A335474, A335516, A335838.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Table[Length[Select[facs[n],Select[Permutations[#],!MatchQ[#,{_,x_,x_,_}]&]!={}&]],{n,100}]

A333487(n) + a(n) = A001055(n).

A335127 A multiset whose multiplicities are the prime indices of n is separable.

Original entry on oeis.org

1, 2, 4, 6, 8, 9, 12, 15, 16, 18, 20, 24, 25, 27, 30, 32, 35, 36, 40, 42, 45, 48, 49, 50, 54, 56, 60, 63, 64, 70, 72, 75, 77, 80, 81, 84, 90, 96, 98, 99, 100, 105, 108, 110, 112, 120, 121, 125, 126, 128, 132, 135, 140, 143, 144, 147, 150, 154, 160, 162, 165

Offset: 1

Gus Wiseman, Jul 02 2020

nonn

A multiset is separable if it has a permutation that is an anti-run, meaning there are no adjacent equal parts.

A multiset whose multiplicities are the prime indices of n (such as row n of A305936) is not the same as the multiset of prime indices of n. 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}.

			The sequence together with the corresponding multisets begins:
   1: {}
   2: {1}
   4: {1,2}
   6: {1,1,2}
   8: {1,2,3}
   9: {1,1,2,2}
  12: {1,1,2,3}
  15: {1,1,1,2,2}
  16: {1,2,3,4}
  18: {1,1,2,2,3}
  20: {1,1,1,2,3}
  24: {1,1,2,3,4}
  25: {1,1,1,2,2,2}
  27: {1,1,2,2,3,3}
  30: {1,1,1,2,2,3}

The complement is A335126.
Anti-run compositions are A003242.
Anti-runs are ranked by A333489.
Separable partitions are A325534.
Inseparable partitions are A325535.
Separable factorizations are A335434.
Inseparable factorizations are A333487.
Separable partitions are ranked by A335433.
Inseparable partitions are ranked by A335448.
Anti-run permutations of prime indices are A335452.
Patterns contiguously matched by compositions are A335457.
Cf. A056239, A106351, A112798, A114938, A292884, A335489, A335516, A335838.

nrmptn[n_]:=Join@@MapIndexed[Table[#2[[1]],{#1}]&,If[n==1,{},Flatten[Cases[FactorInteger[n]//Reverse,{p_,k_}:>Table[PrimePi[p],{k}]]]]];
Select[Range[100],Select[Permutations[nrmptn[#]],!MatchQ[#,{_,x_,x_,_}]&]!={}&]

A335125 Number of anti-run permutations of a multiset whose multiplicities are the prime indices of n.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Jul 01 2020

nonn

A multiset whose multiplicities are the prime indices of n (such as row n of A305936) is not the same as the multiset of prime indices of n. 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}.

An anti-run is a sequence with no adjacent equal parts.

			The a(n) permutations for n = 2, 4, 42, 8, 30, 18:
  (1)  (12)  (1212131)  (123)  (121213)  (12123)
       (21)  (1213121)  (132)  (121231)  (12132)
             (1312121)  (213)  (121312)  (12312)
                        (231)  (121321)  (12321)
                        (312)  (123121)  (13212)
                        (321)  (131212)  (21213)
                               (132121)  (21231)
                               (212131)  (21312)
                               (213121)  (21321)
                               (312121)  (23121)
                                         (31212)
                                         (32121)

Positions of zeros are A335126.
Positions of nonzeros are A335127.
The version for the prime indices themselves is A335452.
Anti-run compositions are A003242.
Anti-runs are ranked by A333489.
Separable partitions are ranked by A335433.
Separable factorizations are A335434.
Inseparable partitions are ranked by A335448.
Patterns contiguously matched by compositions are A335457.
Strict permutations of prime indices are A335489.
Cf. A019472, A056239, A106351, A112798, A114938, A278990, A292884, A325535, A335407, A335463, A335516, A335838.

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]],!MatchQ[#,{_,x_,x_,_}]&]],{n,100}]

A325770 Number of distinct nonempty contiguous subsequences of the integer partition with Heinz number n.

Original entry on oeis.org

0, 1, 1, 2, 1, 3, 1, 3, 2, 3, 1, 5, 1, 3, 3, 4, 1, 5, 1, 5, 3, 3, 1, 7, 2, 3, 3, 5, 1, 6, 1, 5, 3, 3, 3, 8, 1, 3, 3, 7, 1, 6, 1, 5, 5, 3, 1, 9, 2, 5, 3, 5, 1, 7, 3, 7, 3, 3, 1, 9, 1, 3, 5, 6, 3, 6, 1, 5, 3, 6, 1, 11, 1, 3, 5, 5, 3, 6, 1, 9, 4, 3, 1, 9, 3, 3, 3

Offset: 1

Gus Wiseman, May 20 2019

nonn

After a(1) = 0, first differs from A305611 at a(42) = 6, A305611(42) = 7.

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

			The a(84) = 9 distinct nonempty contiguous subsequences of (4,2,1,1) are (1), (2), (4), (1,1), (2,1), (4,2), (2,1,1), (4,2,1), (4,2,1,1).

Cf. A002865, A103295, A112798, A124771, A276024, A325765, A325768, A325769, A335519, A335838.

Table[Length[Union[ReplaceList[If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]],{_,s__,_}:>{s}]]],{n,30}]

a(n) = A335519(n) - 1.

Name corrected by Gus Wiseman, Jun 27 2020

A333487 Number of inseparable factorizations of n into factors > 1.

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 1, 0, 0, 0, 0, 2, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 2, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 5, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 2, 2, 0, 0, 0, 0, 0, 0

Offset: 1

Gus Wiseman, Jul 01 2020

nonn

A multiset is separable if it has a permutation that is an anti-run, meaning there are no adjacent equal parts.

			The a(n) factorizations for n = 4, 16, 96, 144, 64, 192:
  2*2  4*4      2*2*2*12     12*12        8*8          3*4*4*4
       2*2*2*2  2*2*2*2*6    2*2*2*18     4*4*4        2*2*2*24
                2*2*2*2*2*3  2*2*2*2*9    2*2*2*8      2*2*2*2*12
                             2*2*2*2*3*3  2*2*2*2*4    2*2*2*2*2*6
                                          2*2*2*2*2*2  2*2*2*2*3*4
                                                       2*2*2*2*2*2*3

The version for partitions is A325535.
The version for multisets with prescribed multiplicities is A335126.
The separable version is A335434.
Anti-run compositions are A003242.
Anti-runs are ranked by A333489.
Separable partitions are ranked by A335433.
Inseparable partitions are ranked by A335448.
Anti-run permutations of prime indices are A335452.
Patterns contiguously matched by compositions are A335457.
Cf. A106351, A292884, A295370, A333628, A333755, A335463, A335125, A335127, A335407, A335474, A335516, A335838.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Table[Length[Select[facs[n],Select[Permutations[#],!MatchQ[#,{_,x_,x_,_}]&]=={}&]],{n,100}]

a(n) + A335434(n) = A001055(n).

A335516 Number of normal patterns contiguously matched by the prime indices of n in increasing or decreasing order, counting multiplicity.

Original entry on oeis.org

1, 2, 2, 3, 2, 3, 2, 4, 3, 3, 2, 5, 2, 3, 3, 5, 2, 5, 2, 5, 3, 3, 2, 7, 3, 3, 4, 5, 2, 4, 2, 6, 3, 3, 3, 7, 2, 3, 3, 7, 2, 4, 2, 5, 5, 3, 2, 9, 3, 5, 3, 5, 2, 7, 3, 7, 3, 3, 2, 7, 2, 3, 5, 7, 3, 4, 2, 5, 3, 4, 2, 10, 2, 3, 5, 5, 3, 4, 2, 9, 5, 3, 2, 7, 3, 3, 3

Offset: 1

Gus Wiseman, Jun 26 2020

nonn

First differs from A181796 at a(180) = 9, A181796(180) = 10.
First differs from A335549 at a(90) = 7, A335549(90) = 8.
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.
We define a (normal) pattern to be a finite sequence covering an initial interval of positive integers. Patterns are counted by A000670 and ranked by A333217. A sequence S is said to contiguously match a pattern P if there is a contiguous subsequence of S whose parts have the same relative order as P. For example, (3,1,1,3) contiguously matches (1,1,2) and (2,1,1) but not (2,1,2), (1,2,1), (1,2,2), or (2,2,1).

			The a(n) patterns for n = 2, 30, 12, 60, 120, 540, 1500:
  ()   ()     ()     ()      ()       ()        ()
  (1)  (1)    (1)    (1)     (1)      (1)       (1)
       (12)   (11)   (11)    (11)     (11)      (11)
       (123)  (12)   (12)    (12)     (12)      (12)
              (112)  (112)   (111)    (111)     (111)
                     (123)   (112)    (112)     (112)
                     (1123)  (123)    (122)     (122)
                             (1112)   (1112)    (123)
                             (1123)   (1122)    (1123)
                             (11123)  (1222)    (1222)
                                      (11222)   (1233)
                                      (12223)   (11233)
                                      (112223)  (12333)
                                                (112333)

Wikipedia, Permutation pattern
Gus Wiseman, Sequences counting and ranking compositions by the patterns they match or avoid.

The version for standard compositions is A335458.
The not necessarily contiguous version is A335549.
Patterns are counted by A000670 and ranked by A333217.
A number's prime indices are given in the rows of A112798.
Contiguous subsequences of standard compositions are A124771.
Contiguous sub-partitions of prime indices are counted by A335519.
Minimal avoided patterns of prime indices are counted by A335550.
Patterns contiguously matched by partitions are counted by A335838.
Cf. A000005, A056239, A056986, A108917, A124770, A181796, A269134, A333224, A334299, A335457, A335837.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
mstype[q_]:=q/.Table[Union[q][[i]]->i,{i,Length[Union[q]]}];
Table[Length[Union[mstype/@ReplaceList[primeMS[n],{_,s___,_}:>{s}]]],{n,100}]

A337564 Number of sequences of length 2*n covering an initial interval of positive integers and splitting into n maximal runs.

Original entry on oeis.org

1, 1, 6, 80, 1540, 38808, 1206744, 44595408, 1908389340, 92780281880, 5050066185736, 304196411024688, 20087958167374552, 1442953024024996400, 112007566256683719600, 9342904053303870936480, 833388624898522799682780, 79159669418651567937733080

Offset: 0

Gus Wiseman, Sep 03 2020

nonn

Sequences covering an initial interval of positive integers are counted by A000670 and ranked by A333217.

			The a(0) = 1 through a(2) = 6 sequences:
  ()  (1,1)  (1,1,1,2)
             (1,1,2,2)
             (1,2,2,2)
             (2,1,1,1)
             (2,2,1,1)
             (2,2,2,1)
The a(3) = 80 sequences:
  212222  111121  122233  333112  211133
  221222  111211  133222  333211  233111
  222122  112111  222133  112233  331112
  222212  121111  222331  113322  332111
  122221  123333  331222  221133  111223
  211222  133332  332221  223311  111322
  221122  213333  122223  331122  221113
  222112  233331  132222  332211  223111
  112221  333312  222213  112223  311122
  122211  333321  222231  113222  322111
  211122  122333  312222  222113  111123
  221112  133322  322221  222311  111132
  111221  221333  112333  311222  211113
  112211  223331  113332  322211  231111
  122111  333122  211333  111233  311112
  211112  333221  233311  111332  321111

Andrew Howroyd, Table of n, a(n) for n = 0..200

A335461 has this as main diagonal n = 2*k.
A336108 is the version for compositions.
A337504 is the version for compositions and anti-runs.
A337505 is the version for anti-runs.
A000670 counts sequences covering an initial interval.
A005649 counts anti-runs covering an initial interval.
A124767 counts maximal runs in standard compositions.
A333769 gives run lengths in standard compositions.
A337504 counts compositions of 2*n with n maximal anti-runs.
A337565 gives anti-run lengths in standard compositions.
Cf. A001700, A003242, A052841, A060223, A106351, A106356, A269134, A325535, A333489, A333627, A333755, A335838.

allnorm[n_]:=If[n<=0,{{}},Function[s,Array[Count[s,y_/;y<=#]+1&,n]]/@Subsets[Range[n-1]+1]];
Table[Length[Select[Join@@Permutations/@allnorm[2*n],Length[Split[#]]==n&]],{n,0,3}]

\\ here b(n) is A005649.
b(n) = {sum(k=0, n, stirling(n,k,2)*(k + 1)!)}
a(n) = {if(n==0, 1, b(n-1)*binomial(2*n-1,n-1))} \\ Andrew Howroyd, Dec 31 2020

a(n) = A005649(n-1)*binomial(2*n-1,n-1) = A005649(n-1)*A001700(n-1) for n > 0. - Andrew Howroyd, Dec 31 2020

Terms a(5) and beyond from Andrew Howroyd, Dec 31 2020

A335519 Number of contiguous divisors of n.

Original entry on oeis.org

1, 2, 2, 3, 2, 4, 2, 4, 3, 4, 2, 6, 2, 4, 4, 5, 2, 6, 2, 6, 4, 4, 2, 8, 3, 4, 4, 6, 2, 7, 2, 6, 4, 4, 4, 9, 2, 4, 4, 8, 2, 7, 2, 6, 6, 4, 2, 10, 3, 6, 4, 6, 2, 8, 4, 8, 4, 4, 2, 10, 2, 4, 6, 7, 4, 7, 2, 6, 4, 7, 2, 12, 2, 4, 6, 6, 4, 7, 2, 10, 5, 4, 2, 10, 4

Offset: 1

Gus Wiseman, Jun 26 2020

nonn

A divisor of n is contiguous if its prime factors, counting multiplicity, are a contiguous subsequence of the prime factors of n. Explicitly, a divisor d|n is contiguous if n can be written as n = x * d * y where the least prime factor of d is at least the greatest prime factor of x, and the greatest prime factor of d is at most the least prime factor of y.

			The a(84) = 10 distinct contiguous subsequences of (2,2,3,7) are (), (2), (3), (7), (2,2), (2,3), (3,7), (2,2,3), (2,3,7), (2,2,3,7), corresponding to the divisors 1, 2, 3, 7, 4, 6, 21, 12, 42, 84.

Wikipedia, Permutation pattern
Gus Wiseman, Sequences counting and encoding certain classes of multisets
Gus Wiseman, Sequences counting and ranking compositions by the patterns they match or avoid.

The not necessarily contiguous version is A000005.
Each number's prime indices are given in the rows of A112798.
Contiguous subsequences of standard compositions are counted by A124771.
Minimal avoided patterns of prime indices are counted by A335550.
Patterns contiguously matched by partitions are counted by A335838.
Cf. A000670, A056239, A056986, A181796, A325770, A334299, A335457.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Length[Union[ReplaceList[primeMS[n],{_,s___,_}:>{s}]]],{n,100}]

a(n) = A325770(n) + 1.

A335461 Triangle read by rows where T(n,k) is the number of patterns of length n with k runs.

Original entry on oeis.org

1, 0, 1, 0, 1, 2, 0, 1, 4, 8, 0, 1, 6, 24, 44, 0, 1, 8, 48, 176, 308, 0, 1, 10, 80, 440, 1540, 2612, 0, 1, 12, 120, 880, 4620, 15672, 25988, 0, 1, 14, 168, 1540, 10780, 54852, 181916, 296564, 0, 1, 16, 224, 2464, 21560, 146272, 727664, 2372512, 3816548

Offset: 0

Gus Wiseman, Jul 03 2020

We define a pattern to be a finite sequence covering an initial interval of positive integers. Patterns are counted by A000670 and ranked by A333217.

			Triangle begins:
     1
     0     1
     0     1     2
     0     1     4     8
     0     1     6    24    44
     0     1     8    48   176   308
     0     1    10    80   440  1540  2612
     0     1    12   120   880  4620 15672 25988
Row n = 3 counts the following patterns:
  (1,1,1)  (1,1,2)  (1,2,1)
           (1,2,2)  (1,2,3)
           (2,1,1)  (1,3,2)
           (2,2,1)  (2,1,2)
                    (2,1,3)
                    (2,3,1)
                    (3,1,2)
                    (3,2,1)

Andrew Howroyd, Table of n, a(n) for n = 0..1325 (rows 0..50)

Row sums are A000670.
Column n = k is A005649 (anti-run patterns).
Central coefficients are A337564.
The version for compositions is A333755.
Runs of standard compositions are counted by A124767.
Run-lengths of standard compositions are A333769.
Cf. A003242, A052841, A060223, A106351, A106356, A269134, A325535, A333489, A333627, A335838.

allnorm[n_]:=If[n<=0,{{}},Function[s,Array[Count[s,y_/;y<=#]+1&,n]]/@Subsets[Range[n-1]+1]];
Table[Length[Select[Join@@Permutations/@allnorm[n],Length[Split[#]]==k&]],{n,0,5},{k,0,n}]

\\ here b(n) is A005649.
b(n) = {sum(k=0, n, stirling(n,k,2)*(k + 1)!)}
T(n,k)=if(n==0, k==0, b(k-1)*binomial(n-1,k-1)) \\ Andrew Howroyd, Dec 31 2020

T(n,k) = A005649(k-1) * binomial(n-1,k-1) for k > 0. - Andrew Howroyd, Dec 31 2020

cp's OEIS Frontend

A335126 A multiset whose multiplicities are the prime indices of n is inseparable.

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A335434 Number of separable factorizations of n into factors > 1.

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A335127 A multiset whose multiplicities are the prime indices of n is separable.

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A335125 Number of anti-run permutations of a multiset whose multiplicities are the prime indices of n.

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A325770 Number of distinct nonempty contiguous subsequences of the integer partition with Heinz number n.

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A333487 Number of inseparable factorizations of n into factors > 1.

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A335516 Number of normal patterns contiguously matched by the prime indices of n in increasing or decreasing order, counting multiplicity.

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A337564 Number of sequences of length 2*n covering an initial interval of positive integers and splitting into n maximal runs.

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