A335460 -id:A335460 - cp's OEIS frontend

A336107 Number of permutations of the prime indices of n with at least one non-singleton run, or non-separations.

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, 0, 0, 1, 0, 0, 0, 4, 0, 0, 0, 4, 0, 0, 0, 2, 2, 0, 0, 5, 1, 2, 0, 2, 0, 4, 0, 4, 0, 0, 0, 6, 0, 0, 2, 1, 0, 0, 0, 2, 0, 0, 0, 9, 0, 0, 2, 2, 0, 0, 0, 5, 1, 0, 0, 6, 0, 0, 0

Offset: 1

Gus Wiseman, Sep 03 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.

A separation (or Carlitz composition) of a multiset is a permutation with no adjacent equal parts.

			The a(n) non-separations for n = 12, 36, 60, 72, 180, 420:
  (11)  (112)  (1122)  (1123)  (11122)  (11223)  (11234)
        (211)  (1221)  (1132)  (11212)  (11232)  (11243)
               (2112)  (2113)  (11221)  (11322)  (11324)
               (2211)  (2311)  (12112)  (12213)  (11342)
                       (3112)  (12211)  (12231)  (11423)
                       (3211)  (21112)  (13122)  (11432)
                               (21121)  (13221)  (21134)
                               (21211)  (21123)  (21143)
                               (22111)  (21132)  (23114)
                                        (22113)  (23411)
                                        (22131)  (24113)
                                        (22311)  (24311)
                                        (23112)  (31124)
                                        (23211)  (31142)
                                        (31122)  (32114)
                                        (31221)  (32411)
                                        (32112)  (34112)
                                        (32211)  (34211)
                                                 (41123)
                                                 (41132)
                                                 (42113)
                                                 (42311)
                                                 (43112)
                                                 (43211)

A005117 lists positions of zeros, with complement A013929.
A008480 counts permutations of prime indices, ranked by A333221.
A003242 and A335452 count separations, ranked by A333489.
A325535 counts inseparable partitions, ranked by A335448.
A325534 counts separable partitions, ranked by A335433.
Cf. A056239, A112798, A261962, A335451, A335452, A335460, A335489.

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

a(n) = A008480(n) - A335452(n).
a(A000961(n)) = 0 if n is in A027883, otherwise 1.
a(A005117(n)) = 0.
a(n!) = A335459(n).
a(A006939(n)) = A022915(n).

A335520 Number of (1,2,3)-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, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 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, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 3, 0, 0, 0

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).

			The a(n) permutations for n = 30, 60, 120, 210, 180, 480:
  (123)  (1123)  (11123)  (1234)  (11223)  (1111123)
         (1213)  (11213)  (1243)  (11232)  (1111213)
         (1231)  (11231)  (1324)  (12123)  (1111231)
                 (12113)  (1342)  (12132)  (1112113)
                 (12131)  (1423)  (12213)  (1112131)
                 (12311)  (2134)  (12231)  (1112311)
                          (2314)  (12312)  (1121113)
                          (2341)  (12321)  (1121131)
                          (3124)  (21123)  (1121311)
                          (4123)  (21213)  (1123111)
                                  (21231)  (1211113)
                                           (1211131)
                                           (1211311)
                                           (1213111)
                                           (1231111)

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

Positions of nonzero terms are A000977.
These permutations are ranked by A335479.
These compositions are counted by A335514.
Patterns matching this pattern are counted by A335515.
The complement A335521 is the avoiding 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) + A335521(n) = A008480(n).

A335449 Number of (1,2,1)-avoiding permutations of the prime indices of n.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 2, 1, 1, 3, 1, 2, 2, 2, 1, 2, 1, 2, 1, 2, 1, 6, 1, 1, 2, 2, 2, 3, 1, 2, 2, 2, 1, 6, 1, 2, 2, 2, 1, 2, 1, 3, 2, 2, 1, 4, 2, 2, 2, 2, 1, 6, 1, 2, 2, 1, 2, 6, 1, 2, 2, 6, 1, 3, 1, 2, 3, 2, 2, 6, 1, 2, 1, 2, 1, 6, 2, 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 a(n) permutations for n = 2, 10, 36, 54, 324, 30, 1458, 90:
  (1)  (13)  (1122)  (1222)  (112222)  (123)  (1222222)  (1223)
       (31)  (2112)  (2122)  (211222)  (132)  (2122222)  (1322)
             (2211)  (2212)  (221122)  (213)  (2212222)  (2123)
                     (2221)  (222112)  (231)  (2221222)  (2213)
                             (222211)  (312)  (2222122)  (2231)
                                       (321)  (2222212)  (3122)
                                              (2222221)  (3212)
                                                         (3221)

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

The matching version is A335446.
Patterns are counted by A000670.
(1,2,1)-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 counted by 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.
(1,2,1)-avoiding compositions are ranked by A335467.
(1,2,1)-avoiding compositions are counted by A335471.
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

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}]

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.

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).

A335447 Number of (1,2)-matching permutations of the prime indices of n.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 2, 0, 1, 1, 0, 0, 2, 0, 2, 1, 1, 0, 3, 0, 1, 0, 2, 0, 5, 0, 0, 1, 1, 1, 5, 0, 1, 1, 3, 0, 5, 0, 2, 2, 1, 0, 4, 0, 2, 1, 2, 0, 3, 1, 3, 1, 1, 0, 11, 0, 1, 2, 0, 1, 5, 0, 2, 1, 5, 0, 9, 0, 1, 2, 2, 1, 5, 0, 4, 0, 1, 0, 11, 1, 1

Offset: 1

Gus Wiseman, Jun 14 2020

nonn

Depends only on sorted prime signature (A118914).
Also the number of (2,1)-matching permutations of the prime indices of n.
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 = 6, 12, 24, 48, 30, 72, 60:
  (12)  (112)  (1112)  (11112)  (123)  (11122)  (1123)
        (121)  (1121)  (11121)  (132)  (11212)  (1132)
               (1211)  (11211)  (213)  (11221)  (1213)
                       (12111)  (231)  (12112)  (1231)
                                (312)  (12121)  (1312)
                                       (12211)  (1321)
                                       (21112)  (2113)
                                       (21121)  (2131)
                                       (21211)  (2311)
                                                (3112)
                                                (3121)

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

The avoiding version is A000012.
Patterns are counted by A000670.
Positions of zeros are A000961.
(1,2)-matching patterns are counted by A002051.
Permutations of prime indices are counted by A008480.
(1,2)-matching compositions are counted by A056823.
STC-numbers of permutations of prime indices are A333221.
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)-matching compositions are ranked by A335485.
Cf. A056239, A056986, A112798, A181796, A333175, A335451, A335452, A335463, A333175.

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

a(n) = A008480(n) - 1.

A376514 Number of divisors of n that are neither squarefree nor prime powers.

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, 2, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 2, 0, 0, 0, 1, 1, 0, 0, 3, 0, 1, 0, 1, 0, 2, 0, 2, 0, 0, 0, 3, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 5, 0, 0, 1, 1, 0, 0, 0, 3, 0, 0, 0, 3, 0, 0, 0

Offset: 1

Michael De Vlieger, Sep 25 2024

This sequence is distinct from both A062977 and A335460.
A062977(36) = 0 while a(36) = 3 = card({12, 18, 36}).
A335460(60) = 6 while a(60) = 3 = card({12, 20, 60}).

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A000005, A001221, A001222, A027750, A046660, A126706, A376504.

{0}~Join~Table[DivisorSigma[0, n] - Total@ #1 - 2^#2 + #2 & @@ {#, Length[#]} &[FactorInteger[n][[All, -1]] ], {n, 2, 120}] (* or *)
Table[DivisorSum[n, 1 &, Nor[PrimePowerQ[#], SquareFreeQ[#]] &], {n, 120}]

a(n) = tau(n) - 2^omega(n) - bigomega(n) + omega(n).
a(n) = A000005(n) - 2^A001221(n) - A001222(n) + A001221(n).
a(n) = A000005(n) - 2^A001221(n) - A046660(n).
Intersection of row n of A027750 and A126706.
a(n) > 0 for n in A126706.

cp's OEIS Frontend

A336107 Number of permutations of the prime indices of n with at least one non-singleton run, or non-separations.

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A335520 Number of (1,2,3)-matching permutations of the prime indices of n.

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A335449 Number of (1,2,1)-avoiding permutations of the prime indices of n.

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A335450 Number of (2,1,2)-avoiding permutations of the prime indices 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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A335510 Number of (1,1,1)-matching permutations of the prime indices of n.

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

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