A373957 -id:A373957 - cp's OEIS frontend

A386582 Number of distinct inseparable and pairwise disjoint sets of strict integer partitions, one of each exponent in the prime factorization of n.

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

Offset: 1

Gus Wiseman, Jul 31 2025

nonn

A set partition is inseparable iff the underlying set has no permutation whose adjacent elements all belong to different blocks. Note that inseparability only depends on the sizes of the blocks.

			The prime indices of 9216 are {1,1,1,1,1,1,1,1,1,1,2,2}, with a(9216) = 2 choices: {{2},{1,4,5}} and {{2},{1,3,6}}. The other 4 disjoint families {{2},{10}}, {{2},{4,6}}, {{2},{3,7}}, {{2},{1,9}} are separable.
The prime indices of 15552 are {1,1,1,1,1,1,2,2,2,2,2}, with a(15552) = 1 choice: {{5},{1,2,3}}. The other 5 disjoint families {{5},{6}}, {{5},{2,4}}, {{6},{2,3}}, {{6},{1,4}}, {{1,5},{2,3}} are separable.

For separable instead of inseparable we have A386575.
This is the inseparable case of A386587 (ordered version A382525).
Positions of positive terms are A386632.
Positions of first appearances are A386637.
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 (ranks A351294), complement A351293 (ranks A351295).
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, A111133, 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}]

a(2^n) = A111133(n).

A386632 Numbers k such that there is a disjoint inseparable way to choose a strict integer partition of each exponent in the prime factorization of k.

Original entry on oeis.org

8, 16, 27, 32, 64, 81, 125, 128, 243, 256, 343, 512, 625, 729, 1024, 1331, 1536, 2048, 2187, 2197, 2304, 2401, 2560, 3072, 3125, 3456, 3584, 4096, 4608, 4913, 5120, 5184, 5632, 6144, 6400, 6561, 6656, 6859, 6912, 7168, 8192, 8704, 9216, 9728, 10240, 11264

Offset: 1

Gus Wiseman, Aug 04 2025

nonn

First cubefull number (A246549) not in this sequence is 216.
The first term that is not a prime power is 1536.
A set partition is inseparable iff the underlying set has no permutation whose adjacent elements always belong to different blocks. Note that this only depends on the sizes of the blocks.

			The prime indices of 2304 are {1,1,1,1,1,1,1,1,2,2}, and we have disjoint inseparable choice {{4,3,1},{2}}, so 2304 is in the sequence.
The terms together with their prime indices begin:
     8: {1,1,1}
    16: {1,1,1,1}
    27: {2,2,2}
    32: {1,1,1,1,1}
    64: {1,1,1,1,1,1}
    81: {2,2,2,2}
   125: {3,3,3}
   128: {1,1,1,1,1,1,1}
   243: {2,2,2,2,2}
   256: {1,1,1,1,1,1,1,1}
   343: {4,4,4}
   512: {1,1,1,1,1,1,1,1,1}
   625: {3,3,3,3}
   729: {2,2,2,2,2,2}

This is the inseparable case of A351294, positives in A386575, counted by A239455.
Also positions of positive terms in A386582.
A000110 counts set partitions, ordered A000670.
A003242 and A335452 count separations, ranks A333489.
A025065/A386638 counts inseparable type partitions, ranks A335126, sums of A386586.
A325534 counts separable multisets, ranks A335433, sums of A386583.
A325535 counts inseparable multisets, ranks A335448, sums of A386584.
A336106 counts separable type partitions, ranks A335127, sums of A386585.
A386633 counts separable type set partitions, row sums of A386635.
A386634 counts inseparable type set partitions, row sums of A386636.
Cf. A001221, A001222, A051903, A051904, A056239, A130091, A279790, A351293, A373957, A382525, A386587.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
dsj[y_]:=Select[Tuples[IntegerPartitions/@Length/@Split[y]],UnsameQ@@Join@@#&];
insepQ[y_]:=2*Max[y]>Total[y]+1;
Join@@Position[Sign[Table[Length[Select[dsj[prix[n]],insepQ[Length/@#]&]],{n,1000}]],1]

A386578 Irregular triangle read by rows where T(n,k) is the number of permutations of row n of A305936 (a multiset whose multiplicities are the prime indices of n) with k adjacent equal parts.

Original entry on oeis.org

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

Offset: 2

Gus Wiseman, Aug 04 2025

Row 1 is empty, so offset is 2.
Same as A386579 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:
  .  112121  111212  111221  111122  .
     121121  112112  112211  221111
     121211  121112  122111
             211121  211112
             211211
             212111
Triangle begins
   .
   1
   0  1
   2  0
   0  0  1
   1  2  0
   0  0  0  1
   6  0  0
   2  2  2  0
   0  2  2  0
   0  0  0  0  1
   6  6  0  0
   0  0  0  0  0  1
   0  0  3  2  0
   1  4  3  2  0
  24  0  0  0
   0  0  0  0  0  0  1
  12 12  6  0  0
   0  0  0  0  0  0  0  1
   2 12  6  0  0
   0  3  6  4  2  0

Column k = last is A010051.
Row lengths are A056239.
Initial zeros are counted by A252736 = A001222 - 1.
Row sums are A318762.
Column k = 0 is A335125.
For prime indices we have A386577.
Reversing all rows gives 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.
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, A003963, A051903, A051904, A106351, A112798, A238130, A336102, A373957, A374246, A382525, A386581.

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

A374255 Sum of prime factors of n (with multiplicity) minus the greatest possible sum of run-compression of a permutation of the prime factors of n.

Original entry on oeis.org

0, 0, 0, 2, 0, 0, 0, 4, 3, 0, 0, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0, 0, 0, 2, 5, 0, 6, 0, 0, 0, 0, 8, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 4, 7, 0, 0, 0, 0, 3, 0, 2, 0, 0, 0, 0, 0, 0, 0, 10, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 9, 0, 0, 0, 0, 0, 0

Offset: 1

Gus Wiseman, Jul 10 2024

nonn

Contains no ones.

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 96 are {2,2,2,2,2,3}, with sum 13, and we have permutations such as (2,2,2,2,3,2), with run-compression (2,3,2), with sum 7, so a(96) = 13 - 7 = 6.

Positions of first appearances are A280286.
For least instead of greatest sum of run-compression we have A280292.
Positions of zeros are A335433 (separable).
Positions of positive terms are A335448 (inseparable).
For prime indices instead of factors we have A374248.
This is an opposite version of A374250, for prime indices A373956.
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, A246655, A333755, A373954, A374246, A374247.

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

a(n) = A001414(n) - A374250(n).

cp's OEIS Frontend

A386582 Number of distinct inseparable and pairwise disjoint sets of strict integer partitions, one of each exponent in the prime factorization of n.

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A386632 Numbers k such that there is a disjoint inseparable way to choose a strict integer partition of each exponent in the prime factorization of k.

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A386578 Irregular triangle read by rows where T(n,k) is the number of permutations of row n of A305936 (a multiset whose multiplicities are the prime indices of n) with k adjacent equal parts.

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A374255 Sum of prime factors of n (with multiplicity) minus the greatest possible sum of run-compression of a permutation of the prime factors of n.

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