A336104
Number of permutations of the prime indices of A000225(n) = 2^n - 1 with at least one non-singleton run.
Original entry on oeis.org
0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 24, 0, 0, 0, 0, 0, 96, 0, 120, 6, 0, 0, 720, 0, 0, 0, 0, 0, 720, 0, 0, 0, 0, 0, 322560, 0, 0, 0, 5040, 0, 4320, 0, 0, 0, 0, 0, 362880, 0, 0
Offset: 1
The a(21) = 6 permutations of {4, 4, 31, 68}:
(4,4,31,68)
(4,4,68,31)
(31,4,4,68)
(31,68,4,4)
(68,4,4,31)
(68,31,4,4)
A335459 is the version for factorial numbers.
A336105 counts all permutations of this multiset.
A336107 is not restricted to predecessors of powers of 2.
A003242 counts anti-run compositions.
A008480 counts permutations of prime indices.
A333489 ranks anti-run compositions.
A335433 lists numbers whose prime indices have an anti-run permutation.
A335448 lists numbers whose prime indices have no anti-run permutation.
A335452 counts anti-run permutations of prime indices.
A335489 counts strict permutations of prime indices.
The numbers 2^n - 1:
A000225,
A001265,
A001348,
A046051,
A046800,
A046801,
A049093,
A325610,
A325611,
A325612,
A325625.
-
primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Length[Select[Permutations[primeMS[2^n-1]],MatchQ[#,{_,x_,x_,_}]&]],{n,30}]
A350353
Numbers whose multiset of prime factors has a permutation that is not weakly alternating.
Original entry on oeis.org
30, 36, 42, 60, 66, 70, 72, 78, 84, 90, 100, 102, 105, 108, 110, 114, 120, 126, 130, 132, 138, 140, 144, 150, 154, 156, 165, 168, 170, 174, 180, 182, 186, 190, 195, 196, 198, 200, 204, 210, 216, 220, 222, 225, 228, 230, 231, 234, 238, 240, 246, 252, 255, 258
Offset: 1
The terms together with a (generally not unique) non-weakly alternating permutation of each multiset of prime indices begin:
30 : (1,2,3) 100 : (1,3,3,1)
36 : (1,2,2,1) 102 : (1,2,7)
42 : (1,2,4) 105 : (2,3,4)
60 : (1,1,2,3) 108 : (1,2,2,1,2)
66 : (1,2,5) 110 : (1,3,5)
70 : (1,3,4) 114 : (1,2,8)
72 : (1,1,2,2,1) 120 : (1,1,1,2,3)
78 : (1,2,6) 126 : (1,2,4,2)
84 : (1,1,2,4) 130 : (1,3,6)
90 : (1,2,3,2) 132 : (1,1,2,5)
These are the positions of nonzero terms in
A349797.
Below, WA = "weakly alternating":
- WA ordered factorizations are counted by
A349059, complement
A350139.
A008480 counts permutations of prime factors.
A335452 counts anti-run permutations of prime factors, complement
A336107.
A345164 = alternating permutations of prime factors, complement
A350251.
Cf.
A003242,
A335433,
A335448,
A344652,
A344653,
A345171,
A345172,
A345173,
A348379,
A348613,
A349798,
A350252,
A349800.
-
primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
whkQ[y_]:=And@@Table[If[EvenQ[m],y[[m]]<=y[[m+1]],y[[m]]>=y[[m+1]]],{m,1,Length[y]-1}];
Select[Range[100],Select[Permutations[primeMS[#]],!whkQ[#]&&!whkQ[-#]&]!={}&]
Comments