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A350137 Nonsquarefree numbers whose prime signature, except possibly the first and last parts, is all even.

%I A350137 #12 Mar 11 2024 08:30:47
%S A350137 4,8,9,12,16,18,20,24,25,27,28,32,36,40,44,45,48,49,50,52,54,56,63,64,
%T A350137 68,72,75,76,80,81,88,90,92,96,98,99,100,104,108,112,116,117,121,124,
%U A350137 125,126,128,135,136,144,147,148,152,153,160,162,164,169,171,172
%N A350137 Nonsquarefree numbers whose prime signature, except possibly the first and last parts, is all even.
%C A350137 A number's prime signature (row n of A124010) is the sequence of positive exponents in its prime factorization.
%C A350137 Also nonsquarefree numbers whose prime factors, taken in order and with multiplicity, are alternately constant and weakly increasing, starting with either.
%C A350137 Also the Heinz numbers of non-strict integer partitions whose part multiplicities, except possibly the first and last, are all even. These are counted by A349795.
%e A350137 The terms together with their prime indices begin:
%e A350137     4: {1,1}
%e A350137     8: {1,1,1}
%e A350137     9: {2,2}
%e A350137    12: {1,1,2}
%e A350137    16: {1,1,1,1}
%e A350137    18: {1,2,2}
%e A350137    20: {1,1,3}
%e A350137    24: {1,1,1,2}
%e A350137    25: {3,3}
%e A350137    27: {2,2,2}
%e A350137    28: {1,1,4}
%e A350137    32: {1,1,1,1,1}
%e A350137    36: {1,1,2,2}
%e A350137    40: {1,1,1,3}
%e A350137    44: {1,1,5}
%e A350137    45: {2,2,3}
%e A350137    48: {1,1,1,1,2}
%t A350137 Select[Range[100],!SquareFreeQ[#]&&(PrimePowerQ[#]||And@@EvenQ/@Take[Last/@FactorInteger[#],{2,-2}])&]
%Y A350137 This is the nonsquarefree case of the complement of A349794.
%Y A350137 These are the Heinz numbers of the partitions counted by A349795.
%Y A350137 A version for compositions is A349799, counted by A349800.
%Y A350137 A complementary version is A350140, counted by A349796.
%Y A350137 A001250 = alternating permutations, ranked by A349051, complement A348615.
%Y A350137 A005117 = squarefree numbers, complement A013929.
%Y A350137 A025047/A025048/A025049 = alternating compositions, ranked by A345167.
%Y A350137 A056239 adds up prime indices, row sums of A112798, row lengths A001222.
%Y A350137 A124010 = prime signature, sorted A118914.
%Y A350137 A345164 = alternating permutations of prime indices, complement A350251.
%Y A350137 A349052/A129852/A129853 = weakly alternating compositions.
%Y A350137 A349053 = non-weakly alternating compositions, ranked by A349057.
%Y A350137 A349056 = weakly alternating permutations of prime indices.
%Y A350137 A349058 = weakly alternating patterns, complement A350138.
%Y A350137 A349060 = weakly alternating partitions, complement A349061.
%Y A350137 Cf. A000111, A096441, A117298, A335433, A335448, A335452, A344614, A344652, A344653, A345173, A349059, A349797.
%K A350137 nonn
%O A350137 1,1
%A A350137 _Gus Wiseman_, Dec 23 2021