A338869 Shortest most frequent distance among first n primes.

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

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, Log-log 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