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
KEYWORD
nonn
AUTHOR
STATUS
approved