A373957 -id:A373957 - cp's OEIS frontend

A374246 Number of prime factors of n counted with multiplicity (A001222) minus the greatest number of runs possible in a permutation of them (A373957).

0, 0, 0, 1, 0, 0, 0, 2, 1, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 2, 0, 0, 0, 0, 4, 0, 0, 0, 0, 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, 0, 0, 0, 0, 0, 0, 0, 0, 2, 3, 0, 0, 0, 0, 0, 0

Offset: 1

Gus Wiseman, Jul 07 2024

nonn

a(n) = 0 iff n has separable prime factors (A335433). A multiset is separable iff it has a permutation that is an anti-run (meaning there are no adjacent equal parts).

			The runs of the 4 permutations of the prime factors of 24 are:
  ((2,2,2),(3))
  ((2,2),(3),(2))
  ((2),(3),(2,2))
  ((3),(2,2,2))
The longest have length 3, so a(24) = 4 - 3 = 1.

Using the minimum instead of maximum number of runs gives A046660.
Positions of first appearances are A151821 (powers of 2 except 2 itself).
Positions of positive terms are A335448, complement A335433.
This is an opposite version of A373957.
The sister-sequence A374247 uses A001221 instead of A001222.
This is the number of zeros at the end of row n of A374252.
A001221 counts distinct prime factors, A001222 with multiplicity.
A008480 counts permutations of prime factors.
A027746 lists prime factors, row-sums A001414.
A027748 is run-compression of prime factors, row-sums A008472.
A304038 is run-compression of prime indices, row-sums A066328.
A374250 maximizes sum of run-compression, for indices A373956.
Cf. A003242, A007947, A026549, A056239, A124767, A246655, A316413, A333755.

prifacs[n_]:=If[n==1,{}, Flatten[ConstantArray@@@FactorInteger[n]]];
Table[PrimeOmega[n]-Max@@Table[Length[Split[y]], {y,Permutations[prifacs[n]]}],{n,100}]

a(n) = A001222(n) - A373957(n).

A374247 The greatest number of runs possible in a permutation of the prime factors of n (A373957) minus the number of distinct such factors (A001221).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 3, 0, 0, 1, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0

Offset: 1

Gus Wiseman, Jul 07 2024

nonn

If n has separable prime factors (A335433), then a(n) = A001222(n) - A001221(n) = A046660(n). A multiset is separable iff it has an anti-run permutation (meaning there are no adjacent equal parts).

			The runs of the 6 permutations of the prime factors of 36 are:
  ((2,2),(3,3))
  ((2),(3),(2),(3))
  ((2),(3,3),(2))
  ((3),(2,2),(3))
  ((3),(2),(3),(2))
  ((3,3),(2,2))
The longest length is 4, so a(36) = 4 - 2 = 2.

Positions of first appearances appear to be A026549.
Positions of nonzero terms are A126706, complement A303554.
This is an opposite version of A373957.
The sister-sequence A374246 uses A001222 instead of A001221.
This is the number of nonzero terms in row n of A374252.
A003242 counts run-compressed compositions, i.e., anti-runs.
A008480 counts permutations of prime factors, by number of runs A374252.
A027746 lists prime factors, row-sums A001414.
A027748 is run-compression of prime factors, row-sums A008472.
A304038 is run-compression of prime indices, row-sums A066328.
A333755 counts compositions by number of runs.
A335433 lists separable numbers, complement A335448.
A374250 maximizes sum of run-compression, for indices A373956.
Cf. A007947, A046660, A124767, A151821, A238130, A373951, A374251.

prifacs[n_]:=If[n==1,{},Flatten[ConstantArray@@@FactorInteger[n]]];
Table[Max@@Table[Length[Split[y]], {y,Permutations[prifacs[n]]}]-PrimeNu[n],{n,100}]

a(n) = A373957(n) - A001221(n).

A386575 Number of distinct separable and pairwise disjoint sets of strict integer partitions, one of each exponent in the prime factorization of n.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Jul 30 2025

nonn

A set partition is separable iff the underlying set has a permutation whose adjacent elements all belong to different blocks. Note that separability only depends on the sizes of the blocks.

Conjecture: a(n) > 0 iff the multiset of prime factors of n has a permutation with all distinct run lengths.

			The prime indices of 6144 are {1,1,1,1,1,1,1,1,1,1,1,2}, and we have the following a(6144) = 5 choices: {{1},{11}}, {{1},{5,6}}, {{1},{4,7}}, {{1},{3,8}}, {{1},{2,9}}. The other 2 disjoint families {{1},{2,4,5}} and {{1},{2,3,6}} are not separable.
The prime indices of 7776 are {1,1,1,1,1,2,2,2,2,2}, with separable disjoint families {{5},{2,3}}, {{5},{1,4}}, {{1,4},{2,3}}, so a(7776) = 3.
The prime indices of 15552 are {1,1,1,1,1,1,2,2,2,2,2}, with a(15552) = 5 choices: {{5},{6}}, {{5},{2,4}}, {{6},{2,3}}, {{6},{1,4}}, {{1,5},{2,3}}. The other disjoint family {{5},{1,2,3}} is not separable.
The a(n) families for n = 2, 96, 384, 1536, 3456, 20736:
  {{1}}  {{1},{5}}    {{1},{7}}    {{1},{9}}    {{3},{7}}      {{4},{8}}
         {{1},{2,3}}  {{1},{2,5}}  {{1},{2,7}}  {{3},{1,6}}    {{4},{1,7}}
                      {{1},{3,4}}  {{1},{3,6}}  {{3},{2,5}}    {{4},{2,6}}
                                   {{1},{4,5}}  {{7},{1,2}}    {{4},{3,5}}
                                                {{1,2},{3,4}}  {{8},{1,3}}
                                                               {{1,3},{2,6}}

Positions of positive terms are A351294, conjugate A381432.
Positions of 0 are A351295, conjugate A381433.
For inseparable instead of separable we have A386582, see A386632.
This is the separable case of A386587 (ordered version A382525).
A000110 counts set partitions, ordered A000670.
A003242 and A335452 count separations, ranks A333489.
A025065(n - 2) counts partitions of inseparable type, ranks A335126, sums of A386586.
A239455 counts Look-and-Say partitions, complement A351293.
A279790 counts disjoint families on strongly normal multisets.
A325534 counts separable multisets, ranks A335433, sums of A386583.
A325535 counts inseparable multisets, ranks A335448, sums of A386584.
A336106 counts partitions of separable type, ranks A335127, sums of A386585.
A386633 counts separable set partitions, row sums of A386635.
A386634 counts inseparable set partitions, row sums of A386636.
Cf. A001221, A001222, A008480, A051903, A051904, A056239, A130091, A373957, A386580, A386581.

disjointFamilies[y_]:=Union[Sort/@Select[Tuples[IntegerPartitions/@Length/@Split[y]],UnsameQ@@Join@@#&]];
prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
seps[ptn_,fir_]:=If[Total[ptn]==1,{{fir}},Join@@Table[Prepend[#,fir]&/@seps[MapAt[#-1&,ptn,fir],nex],{nex,Select[DeleteCases[Range[Length[ptn]],fir],ptn[[#]]>0&]}]];
seps[ptn_]:=If[Total[ptn]==0,{{}},Join@@(seps[ptn,#]&/@Range[Length[ptn]])];
Table[Length[Select[disjointFamilies[prix[n]],seps[Length/@#]!={}&]],{n,100}]

A374252 Irregular triangle read by rows where T(n,k) is the number of permutations of the prime factors of n with k runs.

Original entry on oeis.org

1, 1, 1, 0, 1, 0, 2, 1, 1, 0, 0, 1, 0, 0, 2, 1, 0, 2, 1, 1, 0, 2, 0, 2, 1, 0, 0, 0, 1, 0, 2, 1, 1, 0, 2, 1, 0, 2, 0, 2, 1, 0, 2, 2, 0, 1, 0, 0, 2, 1, 0, 0, 0, 2, 1, 1, 0, 0, 6, 1, 1, 0, 0, 0, 0, 0, 2, 0, 2, 0, 2, 0, 2, 2, 2, 1, 0, 2, 0, 2, 0, 2, 2, 0, 1

Offset: 2

Gus Wiseman, Jul 07 2024

An alternative form of this sequence (with the same data) has offset 1 and begins with an empty row.

Note that the prime factors of n are separable (A335433) iff the last term of row n is positive.

			The T(36,3) = 2 permutations are (2,3,3,2) and (3,2,2,3).
Row n = 72 counts the following permutations:
  .  (2,2,2,3,3)  (2,2,3,3,2)  (2,2,3,2,3)  (2,3,2,3,2)
     (3,3,2,2,2)  (2,3,3,2,2)  (2,3,2,2,3)
                  (3,2,2,2,3)  (3,2,2,3,2)
                               (3,2,3,2,2)
Triangle begins:
   1:
   2: 1
   3: 1
   4: 1  0
   5: 1
   6: 0  2
   7: 1
   8: 1  0  0
   9: 1  0
  10: 0  2
  11: 1
  12: 0  2  1
  13: 1
  14: 0  2
  15: 0  2
  16: 1  0  0  0
  17: 1
  18: 0  2  1
  19: 1
  20: 0  2  1

Row-lengths are A001222.
Row-sums are A008480 (number of permutations of prime factors).
Column k = 1 is A069513.
For compositions instead of permutations of prime factors we have A238130.
Last column is A335452 (where k = A001222(n)), which counts separations.
Position of the last positive term in row n is A373957(n).
The number of zeros at the end of row n is A374246(n).
The number of nonzero terms in row n is A374247(n).
A001221 counts distinct prime factors.
A003242 counts run-compressed compositions, i.e., anti-runs.
A124767 counts runs in standard compositions, anti-runs A333381.
A333755 counts compositions by number of runs.
A335433 lists separable numbers, complement A335448.
A374250 maximizes sum of run-compression, for indices A373956.
Cf. A026549, A027748, A046660, A126706, A151821, A246655.

prifacs[n_]:=If[n==1,{},Flatten[ConstantArray@@@FactorInteger[n]]];
Table[Length[Select[Permutations[prifacs[n]], Length[Split[#]]==k&]],{n,100},{k,PrimeOmega[n]}]

A374250 Greatest sum of run-compression of a permutation of the prime factors of n.

Original entry on oeis.org

0, 2, 3, 2, 5, 5, 7, 2, 3, 7, 11, 7, 13, 9, 8, 2, 17, 8, 19, 9, 10, 13, 23, 7, 5, 15, 3, 11, 29, 10, 31, 2, 14, 19, 12, 10, 37, 21, 16, 9, 41, 12, 43, 15, 11, 25, 47, 7, 7, 12, 20, 17, 53, 8, 16, 11, 22, 31, 59, 12, 61, 33, 13, 2, 18, 16, 67, 21, 26, 14, 71

Offset: 1

Gus Wiseman, Jul 09 2024

nonn

We define the run-compression of a sequence to be the anti-run obtained by reducing each run of repeated parts to a single part. Alternatively, run-compression removes all parts equal to the part immediately to their left. For example, (1,1,2,2,1) has run-compression (1,2,1).

			The prime factors of 24 are {2,2,2,3}, with permutations such as (2,2,3,2) whose run-compression sums to 7, so a(24) = 7.
The prime factors of 216 are {2,2,2,3,3,3}, with permutations such as (2,3,2,3,2,3) whose run-compression sums to 15, so a(216) = 15.

Positions of 2 are A000079 (powers of two) except 1.
Positions of 3 are A000244 (powers of three) except 1.
For least instead of greatest sum of run-compression we have A008472.
For prime indices instead of factors we have A373956.
For number of runs instead of sum of run-compression we have A373957.
A001221 counts distinct prime factors, A001222 with multiplicity.
A003242 counts run-compressed compositions, i.e., anti-runs.
A007947 (squarefree kernel) represents run-compression of multisets.
A008480 counts permutations of prime factors (or prime indices).
A056239 adds up prime indices, row sums of A112798.
A116861 counts partitions by sum of run-compression.
A304038 lists run-compression of prime indices, sum A066328.
A335433 lists numbers whose prime indices are separable, complement A335448.
A373949 counts compositions by sum of run-compression, opposite A373951.
A374251 run-compresses standard compositions, sum A373953, rank A373948.
Cf. A000040, A001223, A001414, A037201, A116608, A124767, A333755, A373954, A374246, A374247, A374252.

prifacs[n_]:=If[n==1,{},Flatten[ConstantArray@@@FactorInteger[n]]];
Table[Max@@(Total[First/@Split[#]]& /@ Permutations[prifacs[n]]),{n,100}]

a(n) = A001414(n) iff n belongs to A335433 (the separable case, complement A335448), row-sums of A027746.

A373956 Greatest sum of run-compression of a permutation of the prime indices of n.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Jul 06 2024

nonn

We define the run-compression of a sequence to be the anti-run obtained by reducing each run of repeated parts to a single part. Alternatively, run-compression removes all parts equal to the part immediately to their left. For example, (1,1,2,2,1) has run-compression (1,2,1).

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.

			The prime indices of 24 are {1,1,1,2}, with permutations such as (1,1,2,1) whose run-compression sums to 4, so a(24) = 4.
The prime indices of 216 are {1,1,1,2,2,2}, with permutations such as (1,2,1,2,1,2) whose run-compression sums to 9, so a(216) = 9.

Positions of first appearances are 1 followed by the primes A000040.
Positions of 1 are A000079 (powers of two) except 1.
Positions of 2 are A000244 (powers of three) except 1.
Positions of 3 are {6} U A000351 (six or powers of five) except 1.
For number of runs instead of sum of run-compression we have A373957.
For prime factors instead of indices we have A374250.
A001221 counts distinct prime factors, A001222 with multiplicity.
A003242 counts run-compressed compositions, i.e., anti-runs.
A007947 (squarefree kernel) represents run-compression of multisets.
A008480 counts permutations of prime factors (or prime indices).
A056239 adds up prime indices, row sums of A112798.
A116861 counts partitions by sum of run-compression.
A304038 lists run-compression of prime indices, sum A066328.
A335433 lists numbers whose prime indices are separable, complement A335448.
A373949 counts compositions by sum of run-compression, opposite A373951.
A374251 run-compresses standard compositions, sum A373953, rank A373948.
Cf. A001223, A001414, A037201, A116608, A124767, A316413, A333755, A340387, A344415, A373954, A374246, A374247.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Max@@(Total[First/@Split[#]]&/@Permutations[prix[n]]),{n,100}]

a(n) = A056239(n) iff n belongs to A335433 (the separable case), complement A335448.

A386587 Number of ways to choose a pairwise disjoint family of strict integer partitions, one of each exponent in the prime factorization of n.

Original entry on oeis.org

1, 1, 1, 1, 1, 0, 1, 2, 1, 0, 1, 1, 1, 0, 0, 2, 1, 1, 1, 1, 0, 0, 1, 1, 1, 0, 2, 1, 1, 0, 1, 3, 0, 0, 0, 0, 1, 0, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 4, 0, 0, 1, 1, 0, 0, 1, 1, 1, 0, 1, 1, 0, 0, 1, 1, 2, 0, 1, 0, 0, 0, 0

Offset: 1

Gus Wiseman, Aug 06 2025

nonn

First differs from A382525 at a(216) = 1, A382525(216) = 2.

			The prime exponents of 864 = 2^5 * 3^3 are (5,3), with disjoint families {{3},{5}}, {{3},{1,4}}, {{5},{1,2}}, so a(864) = 3.

Positions of positive terms are A351294, conjugate A381432.
Positions of 0 are A351295, conjugate A381433.
For ordered set partitions we have A382525.
Positions of first appearances are A382775.
The separable case is A386575.
The inseparable case is A386582, see A386632.
A000110 counts set partitions, ordered A000670.
A003242 and A335452 count separations, ranks A333489.
A239455 counts Look-and-Say partitions, complement A351293.
A279790 counts disjoint families on strongly normal multisets.
A325534 counts separable multisets, ranks A335433, sums of A386583.
A325535 counts inseparable multisets, ranks A335448, sums of A386584.
A386633 counts separable set partitions, row sums of A386635.
A386634 counts inseparable set partitions, row sums of A386636.
Cf. A001221, A001222, A008480, A025065, A051903, A051904, A056239, A130091, A336106, A373957, A386580.

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

A374248 Sum of prime indices of n (with multiplicity) minus the greatest possible sum of run-compression of a permutation of the prime indices of n.

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 0, 2, 2, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 1, 3, 0, 4, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 2, 4, 0, 0, 0, 0, 2, 0, 1, 0, 0, 0, 0, 0, 0, 0, 5, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 6, 0, 0, 0, 0, 0, 0

Offset: 1

Gus Wiseman, Jul 10 2024

nonn

We define the run-compression of a sequence to be the anti-run obtained by reducing each run of repeated parts to a single part. Alternatively, run-compression removes all parts equal to the part immediately to their left. For example, (1,1,2,2,1) has run-compression (1,2,1).

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.

			The prime indices of 96 are {1,1,1,1,1,2}, with sum 7, and we have permutations such as (1,1,1,1,2,1), with run-compression (1,2,1), with sum 4, so a(96) = 7 - 4 = 3.

Positions of zeros are A335433 (separable).
Positions of positive terms are A335448 (inseparable).
This is an opposite version of A373956, for prime factors A374250.
For prime factors instead of indices we have A374255.
A001221 counts distinct prime factors, A001222 with multiplicity.
A003242 counts run-compressed compositions, i.e., anti-runs.
A007947 (squarefree kernel) represents run-compression of multisets.
A008480 counts permutations of prime factors.
A027746 lists prime factors, row-sums A001414.
A027748 is run-compression of prime factors, row-sums A008472.
A056239 adds up prime indices, row-sums of A112798.
A116861 counts partitions by sum of run-compression.
A304038 is run-compression of prime indices, row-sums A066328.
A373949 counts compositions by sum of run-compression, opposite A373951.
A373957 gives greatest number of runs in a permutation of prime factors.
A374251 run-compresses standard compositions, sum A373953, rank A373948.
A374252 counts permutations of prime factors by number of runs.
Cf. A000040, A026549, A037201, A046660, A124767, A280286, A280292, A333755, A373954, A374246, A374247.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Total[prix[n]]-Max@@(Total[First/@Split[#]]&/@Permutations[prix[n]]),{n,100}]

a(n) = A056239(n) - A373956(n).

A386577 Irregular triangle read by rows where T(n,k) is the number of permutations of the multiset of prime factors of n with k adjacent equal terms.

Original entry on oeis.org

1, 1, 0, 1, 1, 2, 0, 1, 0, 0, 1, 0, 1, 2, 0, 1, 1, 2, 0, 1, 2, 0, 2, 0, 0, 0, 0, 1, 1, 1, 2, 0, 1, 1, 2, 0, 2, 0, 2, 0, 1, 0, 2, 2, 0, 0, 1, 2, 0, 0, 0, 1, 1, 2, 0, 1, 6, 0, 0, 1, 0, 0, 0, 0, 1, 2, 0, 2, 0, 2, 0, 2, 2, 2, 0, 1, 2, 0, 2, 0, 0, 2, 2, 0, 1

Offset: 1

Gus Wiseman, Aug 01 2025

Are the rows all unimodal?

Counts permutations of prime factors by "inseparability". For "separability" we have A374252.

			The prime indices of 12 are {1,1,2}, and we have:
- 1 permutation (1,2,1) with 0 adjacent equal parts
- 2 permutations (1,1,2), (2,1,1) with 1 adjacent equal part
- 0 permutations with 2 adjacent equal parts
so row 12 is (1,2,0).
Row 48 counts the following permutations:
  .  .  (1,1,1,2,1)  (1,1,1,1,2)  .
        (1,1,2,1,1)  (2,1,1,1,1)
        (1,2,1,1,1)
Row 144 counts the following permutations:
  .  (1,1,2,1,2,1)  (1,1,1,2,1,2)  (1,1,1,2,2,1)  (1,1,1,1,2,2)  .
     (1,2,1,1,2,1)  (1,1,2,1,1,2)  (1,1,2,2,1,1)  (2,2,1,1,1,1)
     (1,2,1,2,1,1)  (1,2,1,1,1,2)  (1,2,2,1,1,1)
                    (2,1,1,1,2,1)  (2,1,1,1,1,2)
                    (2,1,1,2,1,1)
                    (2,1,2,1,1,1)
Triangle begins:
   1:
   2: 1
   3: 1
   4: 0  1
   6: 1
   6: 2  0
   7: 1
   8: 0  0  1
   9: 0  1
  10: 2  0
  11: 1
  12: 1  2  0
  13: 1
  14: 2  0
  15: 2  0
  16: 0  0  0  1
  17: 1
  18: 1  2  0
  19: 1
  20: 1  2  0
  21: 2  0
  22: 2  0
  23: 1
  24: 0  2  2  0

Row lengths are A001222.
The minima of each row are A010051.
Sorted positions of first appearances appear to be A025487.
Column k = last is A069513.
Row sums are A168324 or A008480.
The number of trailing zeros in each row is A297155 = A001221-1.
Column k = 1 is A335452.
The number of leading zeros in each row is A374246.
For separability instead of inseparability we have A374252.
For a multiset with prescribed multiplicities we have A386578, separability A386579.
A003242 and A335452 count anti-runs, ranks A333489, patterns A005649.
A025065(n - 2) counts partitions of inseparable type, ranks A335126, sums of A386586.
A124762 gives inseparability of standard compositions, separability A333382.
A325534 counts separable multisets, ranks A335433, sums of A386583.
A325535 counts inseparable multisets, ranks A335448, sums of A386584.
A336106 counts partitions of separable type, ranks A335127, sums of A386585.
Cf. A001221, A051903, A051904, A106351, A238130, A336102, A373957, A382525, A386581, A386632.

Table[Length[Select[Permutations[Flatten[ConstantArray@@@FactorInteger[n]]],Function[q,Length[Select[Range[Length[q]-1],q[[#]]==q[[#+1]]&]]==k]]],{n,30},{k,0,PrimeOmega[n]-1}]

A386579 Number of permutations of row n of A305936 (a multiset whose multiplicities are the prime indices of n) with k adjacent unequal parts.

Original entry on oeis.org

1, 1, 0, 0, 2, 1, 0, 0, 0, 2, 1, 1, 0, 0, 0, 0, 0, 6, 0, 2, 2, 2, 0, 2, 2, 0, 1, 0, 0, 0, 0, 0, 0, 6, 6, 1, 0, 0, 0, 0, 0, 0, 2, 3, 0, 0, 0, 2, 3, 4, 1, 0, 0, 0, 24, 1, 0, 0, 0, 0, 0, 0, 0, 0, 6, 12, 12, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 6, 12, 2, 0, 2, 4, 6, 3, 0

Offset: 2

Gus Wiseman, Aug 04 2025

Row 1 is empty, so offset is 2.
Same as A386578 with rows reversed.
This multiset (row n of A305936) is generally 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}.

			Row n = 21 counts the following permutations:
  .  111122  111221  111212  112121  .
     221111  112211  112112  121121
             122111  121112  121211
             211112  211121
                     211211
                     212111
Triangle begins:
  .
  1
  1  0
  0  2
  1  0  0
  0  2  1
  1  0  0  0
  0  0  6
  0  2  2  2
  0  2  2  0
  1  0  0  0  0
  0  0  6  6
  1  0  0  0  0  0
  0  2  3  0  0
  0  2  3  4  1
  0  0  0 24
  1  0  0  0  0  0  0
  0  0  6 12 12
  1  0  0  0  0  0  0  0
  0  0  6 12  2
  0  2  4  6  3  0

Column k = 0 is A010051.
Row lengths are A056239.
Row sums are A318762.
Column k = last is A335125.
For prime indices we have A374252, reverse A386577.
Reversing all rows gives A386578.
A003242 and A335452 count anti-runs, ranks A333489, patterns A005649.
A025065(n - 2) counts partitions of inseparable type, ranks A335126, sums of A386586.
A124762 gives inseparability of standard compositions, separability A333382.
A305936 is a multiset whose multiplicities are the prime indices of n.
A325534 counts separable multisets, ranks A335433, sums of A386583.
A325535 counts inseparable multisets, ranks A335448, sums of A386584.
A336106 counts partitions of separable type, ranks A335127, sums of A386585.
Cf. A001221, A001222, A003963, A051903, A051904, A106351, A112798, A238130, A336102, A373957, A386581.

nrmptn[n_]:=Join@@MapIndexed[Table[#2[[1]],{#1}]&,If[n==1,{},Flatten[Cases[FactorInteger[n]//Reverse,{p_,k_}:>Table[PrimePi[p],{k}]]]]];
ugt[c_,x_]:=Select[Permutations[c],Function[q,Length[Select[Range[Length[q]-1],q[[#]]!=q[[#+1]]&]]==x]];
Table[Table[Length[ugt[nrmptn[n],k]],{k,0,Length[nrmptn[n]]-1}],{n,30}]

cp's OEIS Frontend

A374246 Number of prime factors of n counted with multiplicity (A001222) minus the greatest number of runs possible in a permutation of them (A373957).

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A374247 The greatest number of runs possible in a permutation of the prime factors of n (A373957) minus the number of distinct such factors (A001221).

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A386575 Number of distinct separable and pairwise disjoint sets of strict integer partitions, one of each exponent in the prime factorization of n.

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A374252 Irregular triangle read by rows where T(n,k) is the number of permutations of the prime factors of n with k runs.

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A374250 Greatest sum of run-compression of a permutation of the prime factors of n.

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A373956 Greatest sum of run-compression of a permutation of the prime indices of n.

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A386587 Number of ways to choose a pairwise disjoint family of strict integer partitions, one of each exponent in the prime factorization of n.

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A374248 Sum of prime indices of n (with multiplicity) minus the greatest possible sum of run-compression of a permutation of the prime indices of n.

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A386577 Irregular triangle read by rows where T(n,k) is the number of permutations of the multiset of prime factors of n with k adjacent equal terms.

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