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A345171 Numbers whose multiset of prime factors has no alternating permutation.

%I A345171 #10 Nov 05 2021 22:18:10
%S A345171 4,8,9,16,24,25,27,32,40,48,49,54,56,64,80,81,88,96,104,112,121,125,
%T A345171 128,135,136,144,152,160,162,169,176,184,189,192,208,224,232,240,243,
%U A345171 248,250,256,270,272,288,289,296,297,304,320,324,328,336,343,344,351
%N A345171 Numbers whose multiset of prime factors has no alternating permutation.
%C A345171 First differs from A335448 in having 270.
%C A345171 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 has the anti-run permutations (2,3,2,1,2) and (2,1,2,3,2).
%C A345171 Also Heinz numbers of integer partitions without a wiggly permutation, where the Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
%e A345171 The sequence of terms together with their prime indices begins:
%e A345171     4: {1,1}
%e A345171     8: {1,1,1}
%e A345171     9: {2,2}
%e A345171    16: {1,1,1,1}
%e A345171    24: {1,1,1,2}
%e A345171    25: {3,3}
%e A345171    27: {2,2,2}
%e A345171    32: {1,1,1,1,1}
%e A345171    40: {1,1,1,3}
%e A345171    48: {1,1,1,1,2}
%e A345171    49: {4,4}
%e A345171    54: {1,2,2,2}
%e A345171    56: {1,1,1,4}
%e A345171    64: {1,1,1,1,1,1}
%e A345171    80: {1,1,1,1,3}
%e A345171    81: {2,2,2,2}
%e A345171    88: {1,1,1,5}
%e A345171    96: {1,1,1,1,1,2}
%t A345171 wigQ[y_]:=Or[Length[y]==0,Length[Split[y]]== Length[y]&&Length[Split[Sign[Differences[y]]]]==Length[y]-1];
%t A345171 Select[Range[100],Select[Permutations[Flatten[ ConstantArray@@@FactorInteger[#]]],wigQ]=={}&]
%Y A345171 Removing squares of primes A001248 gives A344653, counted by A344654.
%Y A345171 A superset of A335448, which is counted by A325535.
%Y A345171 Positions of 0's in A345164.
%Y A345171 The partitions with these Heinz numbers are counted by A345165.
%Y A345171 The complement is A345172, counted by A345170.
%Y A345171 The separable case is A345173, counted by A345166.
%Y A345171 A001250 counts alternating permutations, complement A348615.
%Y A345171 A003242 counts anti-run compositions, complement A261983.
%Y A345171 A025047 counts alternating or wiggly compositions, directed A025048, A025049.
%Y A345171 A325534 counts separable partitions, ranked by A335433.
%Y A345171 A344606 counts alternating permutations of prime indices with twins.
%Y A345171 A344742 ranks twins and partitions with an alternating permutation.
%Y A345171 A345192 counts non-alternating compositions.
%Y A345171 Cf. A001222, A071321, A071322, A316523, A316524, A335126, A344604, A344616, A344652, A344740, A345163, A345168, A345193, A345195, A348380, A348609.
%K A345171 nonn
%O A345171 1,1
%A A345171 _Gus Wiseman_, Jun 13 2021