cp's OEIS Frontend

A350353 Numbers whose multiset of prime factors has a permutation that is not weakly alternating.

%I A350353 #8 Jan 18 2022 05:56:28
%S A350353 30,36,42,60,66,70,72,78,84,90,100,102,105,108,110,114,120,126,130,
%T A350353 132,138,140,144,150,154,156,165,168,170,174,180,182,186,190,195,196,
%U A350353 198,200,204,210,216,220,222,225,228,230,231,234,238,240,246,252,255,258
%N A350353 Numbers whose multiset of prime factors has a permutation that is not weakly alternating.
%C A350353 We define a sequence to be weakly alternating if it is alternately weakly increasing and weakly decreasing, starting with either.
%e A350353 The terms together with a (generally not unique) non-weakly alternating permutation of each multiset of prime indices begin:
%e A350353    30 : (1,2,3)       100 : (1,3,3,1)
%e A350353    36 : (1,2,2,1)     102 : (1,2,7)
%e A350353    42 : (1,2,4)       105 : (2,3,4)
%e A350353    60 : (1,1,2,3)     108 : (1,2,2,1,2)
%e A350353    66 : (1,2,5)       110 : (1,3,5)
%e A350353    70 : (1,3,4)       114 : (1,2,8)
%e A350353    72 : (1,1,2,2,1)   120 : (1,1,1,2,3)
%e A350353    78 : (1,2,6)       126 : (1,2,4,2)
%e A350353    84 : (1,1,2,4)     130 : (1,3,6)
%e A350353    90 : (1,2,3,2)     132 : (1,1,2,5)
%t A350353 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A350353 whkQ[y_]:=And@@Table[If[EvenQ[m],y[[m]]<=y[[m+1]],y[[m]]>=y[[m+1]]],{m,1,Length[y]-1}];
%t A350353 Select[Range[100],Select[Permutations[primeMS[#]],!whkQ[#]&&!whkQ[-#]&]!={}&]
%Y A350353 The strong version is A289553, complement A167171.
%Y A350353 These are the positions of nonzero terms in A349797.
%Y A350353 Below, WA = "weakly alternating":
%Y A350353 - WA compositions are counted by A349052/A129852/A129853.
%Y A350353 - Non-WA compositions are counted by A349053, ranked by A349057.
%Y A350353 - WA permutations of prime factors = A349056, complement A349797.
%Y A350353 - WA patterns are counted by A349058, complement A350138.
%Y A350353 - WA ordered factorizations are counted by A349059, complement A350139.
%Y A350353 - WA partitions are counted by A349060, complement A349061.
%Y A350353 A001250 counts alternating permutations, complement A348615.
%Y A350353 A008480 counts permutations of prime factors.
%Y A350353 A025047 = alternating compositions, ranked by A345167, complement A345192.
%Y A350353 A056239 adds up prime indices, row sums of A112798 (row lengths A001222).
%Y A350353 A071321 gives the alternating sum of prime factors, reverse A071322.
%Y A350353 A335452 counts anti-run permutations of prime factors, complement A336107.
%Y A350353 A345164 = alternating permutations of prime factors, complement A350251.
%Y A350353 Cf. A003242, A335433, A335448, A344652, A344653, A345171, A345172, A345173, A348379, A348613, A349798, A350252, A349800.
%K A350353 nonn
%O A350353 1,1
%A A350353 _Gus Wiseman_, Jan 13 2022