A335451 -id:A335451 - cp's OEIS frontend

A335448 Numbers whose prime indices are inseparable.

4, 8, 9, 16, 24, 25, 27, 32, 40, 48, 49, 54, 56, 64, 80, 81, 88, 96, 104, 112, 121, 125, 128, 135, 136, 144, 152, 160, 162, 169, 176, 184, 189, 192, 208, 224, 232, 240, 243, 248, 250, 256, 272, 288, 289, 296, 297, 304, 320, 324, 328, 336, 343, 344, 351, 352

Offset: 1

Gus Wiseman, Jun 21 2020

nonn

First differs from A212164 in lacking 72.
First differs from A293243 in lacking 72.
No terms are squarefree.
Also Heinz numbers of inseparable partitions (A325535). The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
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.
These are also numbers that can be written as a product of prime numbers, each different from the last but not necessarily different from those prior to the last.
A multiset is inseparable iff its maximal multiplicity is greater than one plus the sum of its remaining multiplicities.

			The sequence of terms together with their prime indices begins:
   4: {1,1}
   8: {1,1,1}
   9: {2,2}
  16: {1,1,1,1}
  24: {1,1,1,2}
  25: {3,3}
  27: {2,2,2}
  32: {1,1,1,1,1}
  40: {1,1,1,3}
  48: {1,1,1,1,2}
  49: {4,4}
  54: {1,2,2,2}
  56: {1,1,1,4}
  64: {1,1,1,1,1,1}
  80: {1,1,1,1,3}
  81: {2,2,2,2}
  88: {1,1,1,5}
  96: {1,1,1,1,1,2}

Complement of A335433.
Separations are counted by A003242 and A335452 and ranked by A333489.
Permutations of prime indices are counted by A008480.
Inseparable partitions are counted by A325535.
Strict permutations of prime indices are counted by A335489.
Cf. A000670, A000961, A005117, A056239, A112798, A181796, A261962, A333221, A335451.

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

A335433 Numbers whose multiset of prime indices is separable.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 26, 28, 29, 30, 31, 33, 34, 35, 36, 37, 38, 39, 41, 42, 43, 44, 45, 46, 47, 50, 51, 52, 53, 55, 57, 58, 59, 60, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 82, 83

Offset: 1

Gus Wiseman, Jul 02 2020

nonn

First differs from A212167 in having 72.
Includes all squarefree numbers A005117.
A multiset is separable if it has a permutation that is an anti-run, meaning there are no adjacent equal parts.
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.
Also Heinz numbers of separable partitions (A325534). The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
Also numbers that cannot be written as a product of prime numbers, each different from the last but not necessarily different from those prior to the last.

			The sequence of terms together with their prime indices begins:
      1: {}          20: {1,1,3}       39: {2,6}
      2: {1}         21: {2,4}         41: {13}
      3: {2}         22: {1,5}         42: {1,2,4}
      5: {3}         23: {9}           43: {14}
      6: {1,2}       26: {1,6}         44: {1,1,5}
      7: {4}         28: {1,1,4}       45: {2,2,3}
     10: {1,3}       29: {10}          46: {1,9}
     11: {5}         30: {1,2,3}       47: {15}
     12: {1,1,2}     31: {11}          50: {1,3,3}
     13: {6}         33: {2,5}         51: {2,7}
     14: {1,4}       34: {1,7}         52: {1,1,6}
     15: {2,3}       35: {3,4}         53: {16}
     17: {7}         36: {1,1,2,2}     55: {3,5}
     18: {1,2,2}     37: {12}          57: {2,8}
     19: {8}         38: {1,8}         58: {1,10}

The version for a multiset with prescribed multiplicities is A335127.
Separable factorizations are counted by A335434.
The complement is A335448.
Separations are counted by A003242 and A335452 and ranked by A333489.
Permutations of prime indices are counted by A008480.
Inseparable partitions are counted by A325535.
Strict permutations of prime indices are counted by A335489.
Cf. A000961, A005117, A056239, A112798, A114938, A181796, A292884, A335407, A335451, A335516.

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

A274174 Number of compositions of n if all summand runs are kept together.

Original entry on oeis.org

1, 1, 2, 4, 7, 12, 22, 36, 60, 97, 162, 254, 406, 628, 974, 1514, 2305, 3492, 5254, 7842, 11598, 17292, 25294, 37090, 53866, 78113, 112224, 161092, 230788, 328352, 466040, 658846, 928132, 1302290, 1821770, 2537156, 3536445, 4897310, 6777806, 9341456, 12858960, 17625970, 24133832, 32910898, 44813228, 60922160, 82569722

Offset: 0

Gregory L. Simay, Jun 12 2016

nonn

a(n^2) is odd. - Gregory L. Simay, Jun 23 2019

Also the number of compositions of n avoiding the patterns (1,2,1) and (2,1,2). - Gus Wiseman, Jul 07 2020

			If the summand runs are blocked together, there are 22 compositions of a(6): 6; 5+1, 1+5, 4+2, 2+4, (3+3), 4+(1+1), (1+1)+4, 1+2+3, 1+3+2, 2+1+3, 2+3+1, 3+1+2, 3+2+1, (2+2+2), 3+(1+1+1), (1+1+1)+3, (2+2)+(1+1), (1+1)+(2+2), 2+(1+1+1+1), (1+1+1+1)+2, (1+1+1+1+1+1).
a(0)=1; a(1)= 1; a(4) = 7; a(9) = 97; a(16) = 2305; a(25) = 78113 and a(36) = 3536445. - _Gregory L. Simay_, Jun 23 2019

Alois P. Heinz, Table of n, a(n) for n = 0..5000
Gus Wiseman, Sequences counting and ranking compositions by the patterns they match or avoid.

Cf. A000070, A101880, A116608.
The version for patterns is A001339.
The version for prime indices is A333175.
The complement (i.e., the matching version) is A335548.
Anti-run compositions are A003242.
(1,2,1)- and (2,1,2)-matching permutations of prime indices are A335462.
(1,2,1)-matching compositions are A335470.
(1,2,1)-avoiding compositions are A335471.
(2,1,2)-matching compositions are A335472.
(2,1,2)-avoiding compositions are A335473.
Cf. A000670, A056986, A181796, A335451, A335452, A335460, A335463.

b:= proc(n, i, p) option remember; `if`(n=0, p!, `if`(i<1, 0,
       add(b(n-i*j, i-1, p+`if`(j=0, 0, 1)), j=0..n/i)))
    end:
a:= n-> b(n$2, 0):
seq(a(n), n=0..50);  # Alois P. Heinz, Jun 12 2016

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],Length[Split[#]]==Length[Union[#]]&]],{n,0,10}] (* Gus Wiseman, Jul 07 2020 *)
b[n_, i_, p_] := b[n, i, p] = If[n == 0, p!, If[i < 1, 0,
    Sum[b[n - i*j, i - 1, p + If[j == 0, 0, 1]], {j, 0, n/i}]]];
a[n_] := b[n, n, 0];
Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Jul 11 2021, after Alois P. Heinz *)

a(n) = Sum_{k>=0} k! * A116608(n,k). - Joerg Arndt, Jun 12 2016

Terms a(9) and beyond from Joerg Arndt, Jun 12 2016

A335452 Number of separations (Carlitz compositions or anti-runs) of the prime indices of n.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Jun 21 2020

nonn

The first term that is not a factorial number is a(180) = 12.
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.
a(n) depends only on the prime signature of n. - Andrew Howroyd, Feb 03 2021

			The a(n) separations for n = 2, 6, 30, 180:
  (1)  (12)  (123)  (12123)
       (21)  (132)  (12132)
             (213)  (12312)
             (231)  (12321)
             (312)  (13212)
             (321)  (21213)
                    (21231)
                    (21312)
                    (21321)
                    (23121)
                    (31212)
                    (32121)

Andrew Howroyd, Table of n, a(n) for n = 1..4096
Wikipedia, Permutation pattern

Separations are counted by A003242 and ranked by A333489.
Patterns are counted by A000670 and ranked by A333217.
Permutations of prime indices are counted by A008480.
Inseparable partitions are counted by A325535 and ranked by A335448.
Cf. A000961, A005117, A056239, A112798, A181796, A261962, A333221, A335451, A335454, A335465, 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}]

F(i, j, r, t) = {sum(k=max(0, i-j), min(min(i,t), (i-j+t)\2), binomial(i, k)*binomial(r-i+1, t+i-j-2*k)*binomial(t-1, k-i+j))}
count(sig)={my(s=vecsum(sig), r=0, v=[1]); for(p=1, #sig, my(t=sig[p]); v=vector(s-r-t+1, j, sum(i=1, #v, v[i]*F(i-1, j-1, r, t))); r += t); v[1]}
a(n)={count(factor(n)[,2])} \\ Andrew Howroyd, Feb 03 2021

A335548 Number of compositions of n with at least one non-contiguous value.

Original entry on oeis.org

0, 0, 0, 0, 1, 4, 10, 28, 68, 159, 350, 770, 1642, 3468, 7218, 14870, 30463, 62044, 125818, 254302, 512690, 1031284, 2071858, 4157214, 8334742, 16699103, 33442208, 66947772, 133986940, 268107104, 536404872, 1073082978, 2146555516, 4293665006, 8588112822

Offset: 0

Gus Wiseman, Jul 08 2020

nonn

Also the number of compositions of n matching the pattern (1,2,1) or (2,1,2).

			The a(4) = 1 through a(6) = 10 compositions:
  (121)  (131)   (141)
         (212)   (1131)
         (1121)  (1212)
         (1211)  (1221)
                 (1311)
                 (2112)
                 (2121)
                 (11121)
                 (11211)
                 (12111)

The complement is A274174.
The version for prime indices is A335460.
Anti-run compositions are A003242.
(1,2,1) and (2,1,2)-matching permutations of prime indices are A335462.
(1,2,1)-matching compositions are A335470.
(1,2,1)-avoiding compositions are A335471.
(2,1,2)-matching compositions are A335472.
(2,1,2)-avoiding compositions are A335473.
Cf. A000670, A001339, A011782, A131044, A261983, A333175, A335451, A335460, A335463.

b:= proc(n, i, p) option remember; `if`(n=0, p!, `if`(i<1, 0,
       add(b(n-i*j, i-1, p+`if`(j=0, 0, 1)), j=0..n/i)))
    end:
a:= n-> ceil(2^(n-1))-b(n$2, 0):
seq(a(n), n=0..50);  # Alois P. Heinz, Jul 09 2020

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],Length[Split[#]]>Length[Union[#]]&]],{n,0,10}]
(* Second program: *)
b[n_, i_, p_] := b[n, i, p] = If[n == 0, p!, If[i<1, 0,
     Sum[b[n-i*j, i-1, p + If[j == 0, 0, 1]], {j, 0, n/i}]]];
a[n_] := Ceiling[2^(n-1)] - b[n, n, 0];
a /@ Range[0, 50] (* Jean-François Alcover, May 21 2021, after Alois P. Heinz *)

a(n) = A011782(n) - A274174(n). - Alois P. Heinz, Jul 09 2020

More terms from Alois P. Heinz, Jul 09 2020

A114938 Number of permutations of the multiset {1,1,2,2,...,n,n} with no two consecutive terms equal.

Original entry on oeis.org

1, 0, 2, 30, 864, 39480, 2631600, 241133760, 29083420800, 4467125013120, 851371260364800, 197158144895712000, 54528028997584665600, 17752366094818747392000, 6720318485119046923315200, 2927066537906697348594432000, 1453437879238150456164433920000

Offset: 0

Hugo Pfoertner, Jan 08 2006

nonn

a(n) is also the number of (0,1)-matrices A=(a_ij) of size n X 2n such that each row has exactly two 1's and each column has exactly one 1 and with the restriction that no 1 stands on the line from a_11 to a_22. - Shanzhen Gao, Feb 24 2010
a(n) is the number of permutations of the multiset {1,1,2,2,...,n,n} with no fixed points. - Alexander Burstein, May 16 2020
Also the number of 2-uniform ordered set partitions of {1...2n} containing no two successive vertices in the same block. - Gus Wiseman, Jul 04 2020

			a(2) = 2 because there are two permutations of {1,1,2,2} avoiding equal consecutive terms: 1212 and 2121.

R. P. Stanley, Enumerative Combinatorics Volume I, Cambridge University Press, 1997. Chapter 2, Sieve Methods, Example 2.2.3, page 68.

Seiichi Manyama, Table of n, a(n) for n = 0..238 (terms 1..100 from Andrew Woods)
H. Eriksson and A. Martin, Enumeration of Carlitz multipermutations, arXiv:1702.04177 [math.CO], 2017.

Cf. A114939 = preferred seating arrangements of n couples.
Cf. A007060 = arrangements of n couples with no adjacent spouses; A007060(n) = 2^n * A114938(n) (this sequence).
Cf. A104548, A193638.
Cf. A278990 = number of loopless linear chord diagrams with n chords.
Cf. A000806 = Bessel polynomial y_n(-1).
The version for multisets with prescribed multiplicities is A335125.
The version for prime indices is A335452.
Anti-run compositions are counted by A003242.
Anti-run compositions are ranked by A333489.
Inseparable partitions are counted by A325535.
Inseparable partitions are ranked by A335448.
Separable partitions are counted by A325534.
Separable partitions are ranked by A335433.
Cf. A007716, A128695, A292884, A335126, A335434, A335451.
Other sequences involving the multiset {1,1,2,2,...,n,n}: A001147, A007717, A020555, A094574, A316972.
Row n=2 of A322093.

[1] cat [n le 2 select 2*(n-1) else n*(2*n-1)*Self(n-1) + (n-1)*n*Self(n-2): n in [1..20]]; // Vincenzo Librandi, Aug 10 2015

Table[Sum[Binomial[n,i](2n-i)!/2^(n-i) (-1)^i,{i,0,n}],{n,0,20}]  (* Geoffrey Critzer, Jan 02 2013, and adapted to the extension by Stefano Spezia, Nov 15 2018 *)
Table[Length[Select[Permutations[Join[Range[n],Range[n]]],!MatchQ[#,{_,x_,x_,_}]&]],{n,0,5}] (* Gus Wiseman, Jul 04 2020 *)

A114938(n)=sum(k=0, n, binomial(n, k)*(-1)^(n-k)*(n+k)!/2^k);
vector(20, n, A114938(n-1)) \\ Michel Marcus, Aug 10 2015

def A114938(n): return (-1)^n*sum(binomial(n,k)*factorial(n+k)//(-2)^k for k in range(n+1))
[A114938(n) for n in range(31)] # G. C. Greubel, Sep 26 2023

a(n) = Sum_{k=0..n} ((binomial(n, k)*(-1)^(n-k)*(n+k)!)/2^k).
a(n) = (-1)^n * n! * A000806(n), n>0. - Vladeta Jovovic, Nov 19 2009
a(n) = n*(2*n-1)*a(n-1) + (n-1)*n*a(n-2). - Vaclav Kotesovec, Aug 07 2013
a(n) ~ 2^(n+1)*n^(2*n)*sqrt(Pi*n)/exp(2*n+1). - Vaclav Kotesovec, Aug 07 2013
a(n) = n! * A278990(n). - Alexander Burstein, May 16 2020
From G. C. Greubel, Sep 26 2023: (Start)
a(n) = (-1)^n * (i/e)*sqrt(2/Pi) * n! * BesselK(n+1/2, -1).
a(n) = [n! * (1/x) * p_{n+1}(x)]|A104548%20for%20p">{x= -1} (See A104548 for p{n}(x)).
E.g.f.: sqrt(Pi/(2*x)) * exp(-(1+x)^2/(2*x)) * erfi((1+x)/sqrt(2*x)).
Sum_{n >= 0} a(n)*x^n/(n!)^2 = exp(sqrt(1-2*x))/sqrt(1-2*x).
Sum_{n >= 0} a(n)*x^n/(n!*(n+1)!) = ( 1 - exp(-1 + sqrt(1-2*x)) )/x. (End)

a(0)=1 prepended by Seiichi Manyama, Nov 15 2018

A333175 If n = Product (p_j^k_j) then a(n) = Sum (a(n/p_j^k_j)), with a(1) = 1.

Original entry on oeis.org

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

Offset: 1

Ilya Gutkovskiy, Mar 11 2020

nonn

Number of ordered prime factorizations of radical of n.

Number of permutations of the prime indices of n (counting multiplicity) avoiding the patterns (1,2,1) and (2,1,2). These are permutations with all equal parts contiguous. Depends only on sorted prime signature (A118914). - Gus Wiseman, Jun 27 2020

			From _Gus Wiseman_, Jun 27 2020 (Start)
The a(n) permutations of prime indices for n = 2, 12, 60:
  (1)  (112)  (1123)
       (211)  (1132)
              (2113)
              (2311)
              (3112)
              (3211)
(End)

Robert Israel, Table of n, a(n) for n = 1..10000
Wikipedia, Permutation pattern
Gus Wiseman, Sequences counting and ranking compositions by the patterns they match or avoid.

Cf. A000142, A000961 (positions of 1's), A001221, A050363, A066504, A069513, A064372, A093320, A292586.
Dominates A335451.
Permutations of prime indices are A008480.
Unsorted prime signature is A124010. Sorted prime signature is A118914.
(1,2,1)-avoiding permutations of prime indices are A335449.
(2,1,2)-avoiding permutations of prime indices are A335450.
(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.
Cf. A056239, A112798, A181796, A333221, A335452, A335463, A335521.

f:= n -> nops(numtheory:-factorset(n))!:
map(f, [$1..100]); # Robert Israel, Mar 12 2020

a[1] = 1; a[n_] := a[n] = Plus @@ (a[n/#[[1]]^#[[2]]] & /@ FactorInteger[n]); Table[a[n], {n, 1, 100}]
a[1] = 1; a[n_] := a[n] = Sum[If[GCD[n/d, d] == 1 && d < n, Boole[PrimePowerQ[n/d]] a[d], 0], {d, Divisors[n]}]; Table[a[n], {n, 1, 100}]
Table[PrimeNu[n]!, {n, 1, 100}]

a(1) = 1; a(n) = Sum_{d|n, d < n, gcd(d, n/d) = 1} A069513(n/d) * a(d).

a(n) = A000142(A001221(n)).

A335460 Number of (1,2,1) or (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, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 2, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 2, 0, 0, 0, 1, 1, 0, 0, 3, 0, 1, 0, 1, 0, 2, 0, 2, 0, 0, 0, 6, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 8, 0, 0, 1, 1, 0, 0, 0, 3, 0, 0, 0, 6, 0, 0, 0

Offset: 1

Gus Wiseman, Jun 20 2020

nonn

Depends 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 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) compositions for n = 12, 24, 48, 36, 60, 72:
  (121)  (1121)  (11121)  (1212)  (1213)  (11212)
         (1211)  (11211)  (1221)  (1231)  (11221)
                 (12111)  (2112)  (1312)  (12112)
                          (2121)  (1321)  (12121)
                                  (2131)  (12211)
                                  (3121)  (21112)
                                          (21121)
                                          (21211)

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

Positions of zeros are A303554.
The (1,2,1)-matching part is A335446.
The (2,1,2)-matching part is A335453.
Replacing "or" with "and" gives A335462.
Permutations of prime indices are counted by A008480.
Unsorted prime signature is A124010. Sorted prime signature is A118914.
STC-numbers of permutations of prime indices are A333221.
(1,2,1) and (2,1,2)-avoiding permutations of prime indices are A333175.
Patterns matched by standard compositions are counted by A335454.
(1,2,1) and (2,1,2)-matching permutations of prime indices are A335462.
Dimensions of downsets of standard compositions are A335465.
Cf. A056239, A056986, A112798, A158005, A181796, A335451, 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}]

A335462 Number of (1,2,1) and (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, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 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, 0, 0, 0, 0, 5, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 1

Gus Wiseman, Jun 20 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 = 36, 72, 270, 144, 300:
  (1,2,1,2)  (1,1,2,1,2)  (2,1,2,3,2)  (1,1,1,2,1,2)  (1,2,3,1,3)
  (2,1,2,1)  (1,2,1,1,2)  (2,1,3,2,2)  (1,1,2,1,1,2)  (1,3,1,2,3)
             (1,2,1,2,1)  (2,2,1,3,2)  (1,1,2,1,2,1)  (1,3,1,3,2)
             (2,1,1,2,1)  (2,2,3,1,2)  (1,2,1,1,1,2)  (1,3,2,1,3)
             (2,1,2,1,1)  (2,3,1,2,2)  (1,2,1,1,2,1)  (1,3,2,3,1)
                          (2,3,2,1,2)  (1,2,1,2,1,1)  (2,1,3,1,3)
                                       (2,1,1,1,2,1)  (2,3,1,3,1)
                                       (2,1,1,2,1,1)  (3,1,2,1,3)
                                       (2,1,2,1,1,1)  (3,1,2,3,1)
                                                      (3,1,3,1,2)
                                                      (3,1,3,2,1)
                                                      (3,2,1,3,1)

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

The avoiding version is A333175.
Replacing "and" with "or" gives A335460.
Positions of nonzero terms are A335463.
Permutations of prime indices are counted by A008480.
Unsorted prime signature is A124010. Sorted prime signature is A118914.
STC-numbers of permutations of prime indices are A333221.
Patterns matched by standard compositions are counted by A335454.
Dimensions of downsets of standard compositions are A335465.
Cf. A056239, A056986, A112798, A158005, A181796, A335451, A335452.

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_,x_,_,y_,_,x_,_}/;x>y]&]],{n,100}]

A335463 Numbers k such that there exists a permutation of the prime indices of k matching both (1,2,1) and (2,1,2).

Original entry on oeis.org

36, 72, 90, 100, 108, 126, 144, 180, 196, 198, 200, 216, 225, 234, 252, 270, 288, 300, 306, 324, 342, 350, 360, 378, 392, 396, 400, 414, 432, 441, 450, 468, 484, 500, 504, 522, 525, 540, 550, 558, 576, 588, 594, 600, 612, 630, 648, 650, 666, 675, 676, 684, 700

Offset: 1

Gus Wiseman, Jun 20 2020

nonn

A prime index of k is a number m such that prime(m) divides k. The multiset of prime indices of k is row k 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 sequence of terms together with their prime indices begins:
   36: {1,1,2,2}
   72: {1,1,1,2,2}
   90: {1,2,2,3}
  100: {1,1,3,3}
  108: {1,1,2,2,2}
  126: {1,2,2,4}
  144: {1,1,1,1,2,2}
  180: {1,1,2,2,3}
  196: {1,1,4,4}
  198: {1,2,2,5}
  200: {1,1,1,3,3}
  216: {1,1,1,2,2,2}
  225: {2,2,3,3}
  234: {1,2,2,6}
  252: {1,1,2,2,4}
  270: {1,2,2,2,3}
  288: {1,1,1,1,1,2,2}
  300: {1,1,2,3,3}

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

Replacing "and" with "or" gives A126706.
Positions of nonzero terms in A335462.
Permutations of prime indices are counted by A008480.
Unsorted prime signature is A124010. Sorted prime signature is A118914.
STC-numbers of permutations of prime indices are A333221.
Patterns matched by standard compositions are counted by A335454.
Cf. A056239, A056986, A112798, A158005, A181796, A333175, A335451, A335452, A335460, A335465.

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

cp's OEIS Frontend

A335448 Numbers whose prime indices are inseparable.

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A335433 Numbers whose multiset of prime indices is separable.

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A274174 Number of compositions of n if all summand runs are kept together.

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A335452 Number of separations (Carlitz compositions or anti-runs) of the prime indices of n.

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A335548 Number of compositions of n with at least one non-contiguous value.

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A114938 Number of permutations of the multiset {1,1,2,2,...,n,n} with no two consecutive terms equal.

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A333175 If n = Product (p_j^k_j) then a(n) = Sum (a(n/p_j^k_j)), with a(1) = 1.

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