

A338869


Shortest most frequent distance among first n primes.


6



1, 1, 2, 2, 2, 2, 2, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 30, 30, 30, 30, 6, 30, 6, 6, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30
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OFFSET

2,3


COMMENTS

Conjecture: Shortest most frequent distance among first n primes is a primorial number (A002110) for n>1.
This sequence is quite related to A338238 with which it shares many terms.
The corresponding frequencies of the most frequent distances among n first primes are in A283371.


LINKS

Table of n, a(n) for n=2..78.
Andres Cicuttin, Loglog plot of the first 2^12 terms


EXAMPLE

For n = 2, the distance between the first two primes 2 and 3 is 1, so the only possible distance is also the most frequent one, then a(2) = 1.
For n = 3, the distances between the first three primes 2, 3 and 5 are 1 = 3  2, 3 = 5  2, and 2 = 5  3, so all three distances are different, have the same frequency, and the shortest among them is 1, then a(3) = 1.
For n = 4, the five different distances between the first four primes 2, 3, 5 and 7 are 1 = 3  2, 2 = 5  3 = 7  5, 3 = 7  4 , 4 = 7  3 and 5 = 7  2, then a(3) = 2 because 2 is the most common distance (two cases) compared with the other distances which appear only once.
For n = 32, the most frequent distances are 30 and 6, and both appear with the same frequency (19 cases), then a(32) = 6 because 6 is the shortest between 30 and 6.


MATHEMATICA

a[n_]:=Module[{pset, p2s, diffp2s, sd, sdgb, sdgbst},
pset=Prime[Range[n]]; (* First n primes *)
p2s=Subsets[pset, {2}]; (* All possible pairs of primes *)
(* Compute all possible distances and the corresponding frequencies *)
diffp2s=Map[Differences, p2s]//Flatten//Tally ;
(* Sort pairs {distance, frequency} by decreasing frequency *)
sd=Sort[diffp2s, #1[[2]]>#2[[2]]&];
(* Gather pairs {dist, freq} with same maximum frequency *)
sdgb=GatherBy[sd, sd[[1]][[2]]==#[[2]] &];
(* Sort selected pairs {dist, freq} with maximum frequency according to increasing distance *)
sdgbst=Sort[sdgb[[1]], #1[[1]]<#2[[1]]&];
(* Finally select and return the minimum distance among those with same maximum frequency *)
sdgbst[[1]][[1]] //Return];
Table[a[n], {n, 2, 100}]


CROSSREFS

Cf. A338238, A002110, A299111, A283371.
Sequence in context: A263407 A221838 A343122 * A104588 A157279 A214080
Adjacent sequences: A338866 A338867 A338868 * A338870 A338871 A338872


KEYWORD

nonn


AUTHOR

Andres Cicuttin, Nov 13 2020


STATUS

approved



