A335452 -id:A335452 - cp's OEIS frontend

A350251 Number of non-alternating permutations of the multiset of prime factors of n.

0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 2, 0, 0, 0, 1, 0, 2, 0, 2, 0, 0, 0, 4, 1, 0, 1, 2, 0, 2, 0, 1, 0, 0, 0, 4, 0, 0, 0, 4, 0, 2, 0, 2, 2, 0, 0, 5, 1, 2, 0, 2, 0, 4, 0, 4, 0, 0, 0, 8, 0, 0, 2, 1, 0, 2, 0, 2, 0, 2, 0, 9, 0, 0, 2, 2, 0, 2, 0, 5, 1, 0, 0, 8, 0, 0, 0

Offset: 1

Gus Wiseman, Jan 08 2022

nonn

A sequence is alternating if it is alternately strictly increasing and strictly decreasing, starting with either. For example, the partition (3,2,2,2,1) has no alternating permutations, even though it does have the anti-run permutations (2,3,2,1,2) and (2,1,2,3,2).

			The a(n) permutations for selected n:
n = 4    12    24     48      60     72      90     96       120
   ----------------------------------------------------------------
    22   223   2223   22223   2235   22233   2335   222223   22235
         322   2232   22232   2253   22323   2353   222232   22253
               2322   22322   2352   22332   2533   222322   22325
               3222   23222   2532   23223   3235   223222   22352
                      32222   3225   23322   3325   232222   22523
                              3522   32223   3352   322222   22532
                              5223   32232   3532            23225
                              5322   32322   5233            23522
                                     33222   5323            25223
                                             5332            25322
                                                             32225
                                                             32252
                                                             32522
                                                             35222
                                                             52223
                                                             52232
                                                             52322
                                                             53222

The non-anti-run case is A336107, complement A335452.
The complement is counted by A345164, with twins A344606.
Positions of nonzero terms are A345171, counted by A345165.
Positions of zeros are A345172, counted by A345170.
Compositions of this type are counted by A345192, ranked by A345168.
Ordered factorizations of this type counted by A348613, complement A348610.
Compositions weakly of this type are counted by A349053, ranked by A349057.
The weak version is A349797, complement A349056.
The case that is also weakly alternating is A349798, compositions A349800.
Patterns of this type are counted by A350252, complement A345194.
A001250 counts alternating permutations, complement A348615.
A003242 counts Carlitz (anti-run) compositions.
A008480 counts permutations of prime factors (ordered prime factorizations).
A025047/A025048/A025049 count alternating compositions, ranked by A345167.
A056239 adds up prime indices, row sums of A112798 (row lengths A001222).
A325534 counts separable partitions, ranked by A335433.
A325535 counts inseparable partitions, ranked by A335448.
A344616 gives the alternating sum of prime indices, reverse A316524.
A349052/A129852/A129853 count weakly alternating compositions.
Cf. A071321, A289553, A316523, A344652, A344653, A344742, A345173, A348379, A349061, A349801, A350138, A350139.

wigQ[y_]:=Or[Length[y]==0,Length[Split[y]] ==Length[y]&&Length[Split[Sign[Differences[y]]]]==Length[y]-1];
Table[Length[Select[Permutations[Flatten[ ConstantArray@@@FactorInteger[n]]],!wigQ[#]&]],{n,100}]

a(n) = A008480(n) - A345164(n).

A335450 Number of (2,1,2)-avoiding permutations of the prime indices of n.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 3, 1, 2, 2, 1, 1, 2, 1, 3, 2, 2, 1, 4, 1, 2, 1, 3, 1, 6, 1, 1, 2, 2, 2, 3, 1, 2, 2, 4, 1, 6, 1, 3, 3, 2, 1, 5, 1, 2, 2, 3, 1, 2, 2, 4, 2, 2, 1, 12, 1, 2, 3, 1, 2, 6, 1, 3, 2, 6, 1, 4, 1, 2, 2, 3, 2, 6, 1, 5, 1, 2, 1, 12, 2, 2

Offset: 1

Gus Wiseman, Jun 14 2020

nonn

Depends only on unsorted prime signature (A124010), but not only on sorted prime signature (A118914).
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 pattern to be a finite sequence covering an initial interval of positive integers. Patterns are counted by A000670. A sequence S is said to match a pattern P if there is a not necessarily contiguous subsequence of S whose parts have the same relative order as P. For example, (3,1,1,3) matches (1,1,2), (2,1,1), and (2,1,2), but avoids (1,2,1), (1,2,2), and (2,2,1).

			The permutations for n = 2, 6, 12, 24, 30, 48, 60, 90:
  (1)  (12)  (112)  (1112)  (123)  (11112)  (1123)  (1223)
       (21)  (211)  (2111)  (132)  (21111)  (1132)  (1322)
                            (213)           (2113)  (2123)
                            (231)           (2311)  (2213)
                            (312)           (3112)  (2231)
                            (321)           (3211)  (3122)
                                                    (3212)
                                                    (3221)

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

Positions of ones are A000961.
Replacing (2,1,2) with (1,2,1) gives A335449.
The matching version is A335453.
Patterns are counted by A000670.
(2,1,2)-avoiding patterns are counted by A001710.
Permutations of prime indices are counted by A008480.
Unsorted prime signature is A124010. Sorted prime signature is A118914.
(1,2,1) and (2,1,2)-avoiding permutations of prime indices are A333175.
STC-numbers of permutations of prime indices are A333221.
(1,2,1) and (2,1,2)-avoiding permutations of prime indices are A335448.
Patterns matched by standard compositions are counted by A335454.
(1,2,1) or (2,1,2)-matching permutations of prime indices are A335460.
(1,2,1) and (2,1,2)-matching permutations of prime indices are A335462.
Dimensions of downsets of standard compositions are A335465.
(2,1,2)-avoiding compositions are ranked by A335469.
(2,1,2)-avoiding compositions are counted by A335473.
(2,2,1)-avoiding compositions are ranked by A335524.
(1,2,2)-avoiding compositions are ranked by A335525.
Cf. A056239, A056986, A112798, A158005, A181796, A335452, A335463.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Length[Select[Permutations[primeMS[n]],!MatchQ[#,{_,x_,_,y_,_,x_,_}/;x>y]&]],{n,100}]

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}]

A335453 Number of (2,1,2)-matching permutations of the prime indices of n.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Jun 14 2020

nonn

Depends only on unsorted prime signature (A124010), but not only on sorted prime signature (A118914).
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 pattern to be a finite sequence covering an initial interval of positive integers. Patterns are counted by A000670. A sequence S is said to match a pattern P if there is a not necessarily contiguous subsequence of S whose parts have the same relative order as P. For example, (3,1,1,3) matches (1,1,2), (2,1,1), and (2,1,2), but avoids (1,2,1), (1,2,2), and (2,2,1).

			The a(n) permutations for n = 18, 36, 54, 72, 90, 108, 144, 180:
  (212)  (1212)  (2122)  (11212)  (2123)  (12122)  (111212)  (12123)
         (2112)  (2212)  (12112)  (2132)  (12212)  (112112)  (12132)
         (2121)          (12121)  (2312)  (21122)  (112121)  (12312)
                         (21112)  (3212)  (21212)  (121112)  (13212)
                         (21121)          (21221)  (121121)  (21123)
                         (21211)          (22112)  (121211)  (21132)
                                          (22121)  (211112)  (21213)
                                                   (211121)  (21231)
                                                   (211211)  (21312)
                                                   (212111)  (21321)
                                                             (23112)
                                                             (23121)
                                                             (31212)
                                                             (32112)
                                                             (32121)

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

References found in the link are not all repeated here.
Positions of ones are A095990.
The avoiding version is A335450.
Replacing (2,1,2) with (1,2,1) gives A335446.
Patterns are counted by A000670.
Permutations of prime indices are counted by A008480.
Unsorted prime signature is A124010. Sorted prime signature is A118914.
(1,2,1) and (2,1,2)-avoiding permutations of prime indices are A333175.
STC-numbers of permutations of prime indices are A333221.
(1,2,1) and (2,1,2)-avoiding permutations of prime indices are A335448.
Patterns matched by standard compositions are counted by A335454.
(1,2,1) or (2,1,2)-matching permutations of prime indices are A335460.
(1,2,1) and (2,1,2)-matching permutations of prime indices are A335462.
Dimensions of downsets of standard compositions are A335465.
(1,2,2)-matching compositions are ranked by A335475.
(2,2,1)-matching compositions are ranked by A335477.
Cf. A056239, A056986, A112798, A158005, A181796, A335452, A335463.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Length[Select[Permutations[primeMS[n]],MatchQ[#,{_,x_,_,y_,_,x_,_}/;x>y]&]],{n,100}]

a(n) + A335450(n) = A008480(n).

A335511 Number of (1,1,1)-avoiding permutations of the prime indices of n.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 0, 1, 2, 1, 3, 1, 2, 2, 0, 1, 3, 1, 3, 2, 2, 1, 0, 1, 2, 0, 3, 1, 6, 1, 0, 2, 2, 2, 6, 1, 2, 2, 0, 1, 6, 1, 3, 3, 2, 1, 0, 1, 3, 2, 3, 1, 0, 2, 0, 2, 2, 1, 12, 1, 2, 3, 0, 2, 6, 1, 3, 2, 6, 1, 0, 1, 2, 3, 3, 2, 6, 1, 0, 0, 2, 1, 12, 2, 2

Offset: 1

Gus Wiseman, Jun 19 2020

nonn

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. A sequence S is said to match a pattern P if there is a not necessarily contiguous subsequence of S whose parts have the same relative order as P. For example, (3,1,1,3) matches (1,1,2), (2,1,1), and (2,1,2), but avoids (1,2,1), (1,2,2), and (2,2,1).

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

Patterns avoiding this pattern are counted by A080599.
These compositions are counted by A232432.
The (1,1)-avoiding version is A335451.
The complement A335510 is the matching version.
These permutations are ranked by A335513.
Patterns are counted by A000670 and ranked by A333217.
Permutations of prime indices are counted by A008480.
Anti-run permutations of prime indices are counted by A335452.
Cf. A056239, A056986, A112798, A238279, A281188, A333221, A333755, A335456, A335460, A335462, A335463.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Length[Select[Permutations[primeMS[n]],!MatchQ[#,{_,x_,_,x_,_,x_,_}]&]],{n,100}]

If n is cubefree, a(n) = A008480(n), otherwise a(n) = 0.

A335550 Number of minimal normal patterns avoided by the prime indices of n in increasing or decreasing order, counting multiplicity.

Original entry on oeis.org

1, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4, 3, 3, 3, 3, 3, 4, 3, 4, 3, 3, 3, 4, 3, 3, 3, 4, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4, 3, 3, 3, 4, 4, 3, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 3, 3, 4, 3, 3, 4, 3, 3, 3, 3, 4, 3, 3

Offset: 1

Gus Wiseman, Jun 26 2020

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 match a pattern P if there is a not necessarily contiguous subsequence of S whose parts have the same relative order as P. For example, (3,1,1,3) matches (1,1,2), (2,1,1), and (2,1,2), but avoids (1,2,1), (1,2,2), and (2,2,1).

			The a(12) = 4 minimal patterns avoiding (1,1,2) are: (2,1), (1,1,1), (1,2,2), (1,2,3).
The a(30) = 3 minimal patterns avoiding (1,2,3) are: (1,1), (2,1), (1,2,3,4).

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

The version for standard compositions is A335465.
Patterns are counted by A000670.
Sum of prime indices is A056239.
Each number's prime indices are given in the rows of A112798.
Patterns are ranked by A333217.
Patterns matched by compositions are counted by A335456.
Patterns matched by prime indices are counted by A335549.
Patterns matched by partitions are counted by A335837.
Cf. A124770, A124771, A181796, A269134, A299702, A333257, A335452, A335516.

It appears that for n > 1, a(n) = 3 if n is a power of a squarefree number (A072774), and a(n) = 4 otherwise.

A386638 Number of integer partitions of n of inseparable type.

Original entry on oeis.org

0, 0, 1, 1, 2, 2, 4, 4, 7, 7, 12, 12, 19, 19, 30, 30, 45, 45, 67, 67, 97, 97, 139, 139, 195, 195, 272, 272, 373, 373, 508, 508, 684, 684, 915, 915, 1212, 1212, 1597, 1597, 2087, 2087, 2714, 2714, 3506, 3506, 4508, 4508, 5763, 5763, 7338, 7338, 9296, 9296

Offset: 0

Gus Wiseman, Aug 14 2025

nonn

A multiset is inseparable iff it has no permutation without adjacent equal parts. It is of inseparable type iff any multiset with those multiplicities (type) is inseparable. For example, {1,1,2} is separable but {1,1,1,2} is not; hence (2,1) is of separable type but (3,1) is not.

Also the number of integer partitions of n whose greatest part is at least two more than the sum of all the other parts.

			The a(2) = 1 through a(10) = 12 partitions (A=10):
  (2)  (3)  (4)   (5)   (6)    (7)    (8)     (9)     (A)
            (31)  (41)  (42)   (52)   (53)    (63)    (64)
                        (51)   (61)   (62)    (72)    (73)
                        (411)  (511)  (71)    (81)    (82)
                                      (521)   (621)   (91)
                                      (611)   (711)   (622)
                                      (5111)  (6111)  (631)
                                                      (721)
                                                      (811)
                                                      (6211)
                                                      (7111)
                                                      (61111)

Reduplication of A000070 shifted right.
Same as A025065 shifted right twice.
The Heinz numbers of these partitions are A335126.
Row sums of A386586.
A003242 and A335452 count anti-runs, ranks A333489, patterns A005649.
A239455 counts Look-and-Say partitions, inseparable case A386632.
A325534 counts separable multisets, ranks A335433, sums of A386583.
A325535 counts inseparable multisets, ranks A335448, sums of A386584.
A335434 counts separable factorizations, inseparable A333487.
A336103 counts normal separable multisets, inseparable A336102.
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. A000110, A008284, A047966, A072233, A106351, A124762, A217605, A239964, A279790, A335125, A351293, A386577.

Table[Length[Select[IntegerPartitions[n],2*Max@@#>1+n&]],{n,0,15}]

For n>1, a(n) = A025065(n-2).

a(n) = A000041(n) - A336106(n).

A335510 Number of (1,1,1)-matching permutations of the prime indices of n.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 5, 0, 0, 0, 0, 0, 4, 0, 4, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 10, 0, 0, 0, 0, 0, 0, 0, 5, 1, 0, 0, 0, 0, 0, 0

Offset: 1

Gus Wiseman, Jun 19 2020

nonn

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 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 match a pattern P if there is a not necessarily contiguous subsequence of S whose parts have the same relative order as P. For example, (3,1,1,3) matches (1,1,2), (2,1,1), and (2,1,2), but avoids (1,2,1), (1,2,2), and (2,2,1).

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

Patterns matching this pattern are counted by A335508.
These compositions are counted by A335455.
The (1,1)-matching version is A335487.
The complement A335511 is the avoiding version.
These permutations are ranked by A335512.
Permutations of prime indices are counted by A008480.
Patterns are counted by A000670 and ranked by A333217.
Anti-run permutations of prime indices are counted by A335452.
Cf. A056239, A056986, A112798, A238279, A281188, A333221, A333755, A335456, A335460, A335462, A335463.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Length[Select[Permutations[primeMS[n]],MatchQ[#,{_,x_,_,x_,_,x_,_}]&]],{n,0,100}]

If n is cubefree, a(n) = 0; otherwise a(n) = A008480(n).

A335521 Number of (1,2,3)-avoiding permutations of the prime indices of n.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Jun 19 2020

nonn

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 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 match a pattern P if there is a not necessarily contiguous subsequence of S whose parts have the same relative order as P. For example, (3,1,1,3) matches (1,1,2), (2,1,1), and (2,1,2), but avoids (1,2,1), (1,2,2), and (2,2,1).

			The a(n) permutations for n = 1, 6, 12, 24, 30, 36, 60, 72, 120:
  ()  (12)  (112)  (1112)  (132)  (1122)  (1132)  (11122)  (11132)
      (21)  (121)  (1121)  (213)  (1212)  (1312)  (11212)  (11312)
            (211)  (1211)  (231)  (1221)  (1321)  (11221)  (11321)
                   (2111)  (312)  (2112)  (2113)  (12112)  (13112)
                           (321)  (2121)  (2131)  (12121)  (13121)
                                  (2211)  (2311)  (12211)  (13211)
                                          (3112)  (21112)  (21113)
                                          (3121)  (21121)  (21131)
                                          (3211)  (21211)  (21311)
                                                  (22111)  (23111)
                                                           (31112)
                                                           (31121)
                                                           (31211)
                                                           (32111)

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

These compositions are counted by A102726.
Patterns avoiding this pattern are counted by A226316.
The complement A335520 is the matching version.
Permutations of prime indices are counted by A008480.
Patterns are counted by A000670 and ranked by A333217.
Anti-run permutations of prime indices are counted by A335452.
Cf. A056239, A056986, A112798, A238279, A281188, A333221, A333755, A335456, A335460, A335462, A335463.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Length[Select[Permutations[primeMS[n]],!MatchQ[#,{_,x_,_,y_,_,z_,_}/;x

For n > 0, a(n) + A335520(n) = A008480(n).

A350137 Nonsquarefree numbers whose prime signature, except possibly the first and last parts, is all even.

Original entry on oeis.org

4, 8, 9, 12, 16, 18, 20, 24, 25, 27, 28, 32, 36, 40, 44, 45, 48, 49, 50, 52, 54, 56, 63, 64, 68, 72, 75, 76, 80, 81, 88, 90, 92, 96, 98, 99, 100, 104, 108, 112, 116, 117, 121, 124, 125, 126, 128, 135, 136, 144, 147, 148, 152, 153, 160, 162, 164, 169, 171, 172

Offset: 1

Gus Wiseman, Dec 23 2021

nonn

A number's prime signature (row n of A124010) is the sequence of positive exponents in its prime factorization.
Also nonsquarefree numbers whose prime factors, taken in order and with multiplicity, are alternately constant and weakly increasing, starting with either.
Also the Heinz numbers of non-strict integer partitions whose part multiplicities, except possibly the first and last, are all even. These are counted by A349795.

			The terms together with their prime indices begin:
    4: {1,1}
    8: {1,1,1}
    9: {2,2}
   12: {1,1,2}
   16: {1,1,1,1}
   18: {1,2,2}
   20: {1,1,3}
   24: {1,1,1,2}
   25: {3,3}
   27: {2,2,2}
   28: {1,1,4}
   32: {1,1,1,1,1}
   36: {1,1,2,2}
   40: {1,1,1,3}
   44: {1,1,5}
   45: {2,2,3}
   48: {1,1,1,1,2}

This is the nonsquarefree case of the complement of A349794.
These are the Heinz numbers of the partitions counted by A349795.
A version for compositions is A349799, counted by A349800.
A complementary version is A350140, counted by A349796.
A001250 = alternating permutations, ranked by A349051, complement A348615.
A005117 = squarefree numbers, complement A013929.
A025047/A025048/A025049 = alternating compositions, ranked by A345167.
A056239 adds up prime indices, row sums of A112798, row lengths A001222.
A124010 = prime signature, sorted A118914.
A345164 = alternating permutations of prime indices, complement A350251.
A349052/A129852/A129853 = weakly alternating compositions.
A349053 = non-weakly alternating compositions, ranked by A349057.
A349056 = weakly alternating permutations of prime indices.
A349058 = weakly alternating patterns, complement A350138.
A349060 = weakly alternating partitions, complement A349061.
Cf. A000111, A096441, A117298, A335433, A335448, A335452, A344614, A344652, A344653, A345173, A349059, A349797.

Select[Range[100],!SquareFreeQ[#]&&(PrimePowerQ[#]||And@@EvenQ/@Take[Last/@FactorInteger[#],{2,-2}])&]

cp's OEIS Frontend

A350251 Number of non-alternating permutations of the multiset of prime factors of n.

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A335450 Number of (2,1,2)-avoiding permutations 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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A335453 Number of (2,1,2)-matching permutations of the prime indices of n.

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A335511 Number of (1,1,1)-avoiding permutations of the prime indices of n.

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A335550 Number of minimal normal patterns avoided by the prime indices of n in increasing or decreasing order, counting multiplicity.

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A386638 Number of integer partitions of n of inseparable type.

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A335510 Number of (1,1,1)-matching permutations of the prime indices of n.

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