login
The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

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