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A349794
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Numbers whose prime signature has an odd term other than the first or last.
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11
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30, 42, 60, 66, 70, 78, 84, 102, 105, 110, 114, 120, 130, 132, 138, 140, 150, 154, 156, 165, 168, 170, 174, 182, 186, 190, 195, 204, 210, 220, 222, 228, 230, 231, 238, 240, 246, 255, 258, 260, 264, 266, 270, 273, 276, 280, 282, 285, 286, 290, 294, 300, 308
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OFFSET
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1,1
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COMMENTS
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A number's prime signature (row n of A124010) is the sequence of positive exponents in its prime factorization.
Also numbers whose multiset of prime factors is not weakly alternating, where we define a sequence to be weakly alternating if it is alternately weakly increasing and weakly decreasing, starting with either. This sequence looks at the somewhat degenerate case where no strict decreases are allowed.
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LINKS
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EXAMPLE
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The terms and their prime indices begin:
30: {1,2,3}
42: {1,2,4}
60: {1,1,2,3}
66: {1,2,5}
70: {1,3,4}
78: {1,2,6}
84: {1,1,2,4}
102: {1,2,7}
105: {2,3,4}
110: {1,3,5}
114: {1,2,8}
120: {1,1,1,2,3}
130: {1,3,6}
132: {1,1,2,5}
138: {1,2,9}
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MATHEMATICA
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Select[Range[100], PrimeNu[#]>1&&!And@@EvenQ/@Take[Last/@FactorInteger[#], {2, -2}]&]
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CROSSREFS
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The non-strict case is counted by A349795.
A096441 counts weakly alternating partitions if 0 is appended.
A345164 counts alternating permutations of prime indices, weak A349056.
A345170 counts partitions w/ an alternating permutation, ranked by A345172.
A349052 counts weakly alternating compositions.
A349059 counts weakly alternating ordered factorizations, strong A348610.
Cf. A003242, A112798, A114901, A117298, A128761, A129852, A129853, A344614, A344615, A345165, A349796, A349798.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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