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 A334238 Rows n in A334184 that are not unimodal. 2
 57, 63, 171, 258, 266, 294, 301, 329, 342, 343, 354, 361, 377, 378, 379, 381, 387, 399, 423, 437, 441, 462, 463, 469, 474, 481, 483, 489, 506, 513, 529, 567, 603, 621, 642, 643, 689, 798, 817, 889, 903, 931, 978, 1026, 1083, 1141, 1143, 1161, 1169, 1197, 1204 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS Consider the mappings k -> (k - (k/p)), across primes p | k. a(n) = rank levels of antichains in the poset resulting from taking distinct terms generated by the mapping and preserving the order of their generation. We deem a series of rank levels, such as those of n = 15, i.e., row 15 of A334184 = [1, 2, 3, 2, 1, 1], as unimodal, as the terms increase to a point, then decrease. Early terms may suggest that 2^i +/- 1 appear often in a(n). Given 10000 terms, the only such instances are {63, 513, 2047, 16383} for i = {6, 9, 11, 14}. a(n) for 1 <= n <= 710 are bimodal. Are there rows n > 710 in A334184 that increase and decrease more than twice? LINKS Peter Kagey, Table of n, a(n) for n = 1..10000 Michael De Vlieger Hasse diagrams of the 24 least terms of this sequence. EXAMPLE Example: n = 57 is the smallest number for which rank levels of antichains is not unimodal, under the poset formed from distinct terms resulting from the mapping f(n) := n -> n - n/p across primes p | n.     Hasse diagram     Row 57 of A334184     -------------     -----------------         57            1         | \         |  \         54  38        2         | \/  \         | /\   \         36  27  19    3         | \ |  /         |  \| /        24   18        2        /|  /|       / | / |     16  12  9         3      | /|  /      |/ |_/      8  6             2      | /|      |/ |      4  3             2      | /      |/      2                1      |      |      1                1 MATHEMATICA Select[Range[2, 600], Function[k, Which[IntegerQ@ Log2@ k, False, And[PrimeQ@ k, IntegerQ@ Log2[k - 1]], False, True, ! AllTrue[Drop[#,  FirstPosition[#, _?(# < 0 &)][[1]] - 1 ], # <= 0 &] &@ Sign@ Differences@ Map[Length@ Union@ # &, Transpose@ If[k == 1, {{1}}, NestWhile[If[Length[#] == 0, Map[{k, #} &, # - # /FactorInteger[#][[All, 1]] ], Union[Join @@  Map[Function[{w, n}, Map[Append[w, If[n == 0, 0, n - n/#]] &, FactorInteger[n][[All, 1]] ]] @@ {#, Last@ #} &, #]] ] &, k, If[ListQ[#], AllTrue[#, Last[#] > 1 &], # > 1] &]]]]]] CROSSREFS Cf. A334184. Sequence in context: A042623 A072466 A216183 * A336328 A056082 A218562 Adjacent sequences:  A334235 A334236 A334237 * A334239 A334240 A334241 KEYWORD nonn AUTHOR Michael De Vlieger, Peter Kagey, Antti Karttunen, Apr 19 2020 STATUS approved

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Last modified August 8 19:29 EDT 2020. Contains 336298 sequences. (Running on oeis4.)