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A350140
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Nonsquarefree numbers whose prime signature has at least one odd part other the first or last.
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5
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60, 84, 120, 132, 140, 150, 156, 168, 204, 220, 228, 240, 260, 264, 270, 276, 280, 294, 300, 308, 312, 315, 336, 340, 348, 364, 372, 378, 380, 408, 420, 440, 444, 456, 460, 476, 480, 490, 492, 495, 516, 520, 528, 532, 540, 552, 560, 564, 572, 580, 585, 588
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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 Heinz numbers of non-weakly alternating non-strict integer partitions, where we define a sequence to be weakly alternating if it is alternately weakly increasing and weakly decreasing, starting with either. These partitions are counted by A349796. This sequence involves the somewhat degenerate case where no strict increases are allowed.
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LINKS
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FORMULA
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EXAMPLE
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The terms together with their Heinz partitions begin (A-E = 10-14):
60: (3211) 276: (9211) 420: (43211)
84: (4211) 280: (43111) 440: (53111)
120: (32111) 294: (4421) 444: (C211)
132: (5211) 300: (33211) 456: (82111)
140: (4311) 308: (5411) 460: (9311)
150: (3321) 312: (62111) 476: (7411)
156: (6211) 315: (4322) 480: (3211111)
168: (42111) 336: (421111) 490: (4431)
204: (7211) 340: (7311) 492: (D211)
220: (5311) 348: (A211) 495: (5322)
228: (8211) 364: (6411) 516: (E211)
240: (321111) 372: (B211) 520: (63111)
260: (6311) 378: (42221) 528: (521111)
264: (52111) 380: (8311) 532: (8411)
270: (32221) 408: (72111) 540: (322211)
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MATHEMATICA
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Select[Range[300], !SquareFreeQ[#]&&PrimeNu[#]>1&& !And@@EvenQ/@Take[Last/@FactorInteger[#], {2, -2}]&]
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CROSSREFS
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Including all nonsquarefree numbers gives A013929, complement A005117.
The strict instead of non-strict version is A336568, counted by A347548.
A version for compositions allowing strict is A349057, counted by A349053.
These partitions are counted by A349796.
The complement in nonsquarefree partitions is A350137, counted by A349795.
A003242 = Carlitz (anti-run) compositions.
A096441 = weakly alternating 0-appended partitions.
A345164 = alternating permutations of prime indices, complement A350251.
A345170 = partitions w/ an alternating permutation, ranked by A345172.
A349056 = weakly alternating permutations of prime indices.
A349798 = weakly but not strongly alternating perms of prime indices.
Cf. A000111, A047967, A333213, A335448, A344615, A344653, A345173, A349054, A349059, A349797, A349799.
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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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