A338238 Minimum number of rotations for a second maximum cyclic autocorrelation of the first n terms of the characteristic function of primes.

1, 1, 1, 2, 3, 2, 2, 2, 2, 2, 4, 2, 6, 2, 8, 2, 4, 2, 6, 6, 6, 6, 6, 4, 6, 6, 6, 6, 6, 2, 6, 6, 6, 6, 12, 6, 6, 6, 6, 6, 18, 6, 6, 6, 6, 6, 24, 6, 6, 6, 6, 6, 24, 6, 6, 6, 24, 6, 6, 6, 6, 6, 6, 6, 24, 6, 6, 6, 6, 6, 24, 6, 6, 6, 6, 6, 24, 6, 6, 6, 6, 6, 30, 6, 30, 6, 12, 6, 30, 6, 6, 6, 6, 6, 30, 6, 30

Offset

2,4

Comments

It seems that most frequent terms among the first ones assume values 1, 2, 6, 30, 210, 2310, . . . Primorials? Several scatter plots of sequences of different lengths suggest this pattern (See Link).

Links

Table of n, a(n) for n=2..98.

Andres Cicuttin, Several scatter plots of sequences of different lengths

Example

The primes among the first 5 positive integers (1,2,3,4,5) are 2, 3, and 5, then the corresponding characteristic function of primes is (0,1,1,0,1) (See A010051) and the corresponding five possible cyclic autocorrelations are the dot products between (0,1,1,0,1) and its rotations as shown here below:
(0,1,1,0,1).(0,1,1,0,1) = 0*0 + 1*1 + 1*1 + 0*0 + 1*1 = 3, (0 rotations)
(0,1,1,0,1).(1,0,1,1,0) = 0*1 + 1*0 + 1*1 + 0*1 + 1*0 = 1, (1 rotation)
(0,1,1,0,1).(0,1,0,1,1) = 0*0 + 1*1 + 1*0 + 0*1 + 1*1 = 2, (2 rotations)
(0,1,1,0,1).(1,0,1,0,1) = 0*1 + 1*0 + 1*1 + 0*0 + 1*1 = 2, (3 rotations)
(0,1,1,0,1).(1,1,0,1,0) = 0*1 + 1*1 + 1*0 + 0*1 + 1*0 = 1, (4 rotations)
The maximum value of the cyclic autocorrelation is always trivially obtained with zero rotations. In this example, the maximum value is 3 and the second maximum is 2, then a(5)=2 because it is needed a minimum of 2 rotations to obtain the second maximum.

Mathematica

nmax = 2^7;
b = Table[If[PrimeQ[i], 1, 0], {i, 1, nmax}];
tab = Table[Table[b[[1;; n]].RotateRight[b[[1;; n]], j], {j, 1, n-1}], {n, 2, nmax}];
tabmaxs = Table[Max[tab[[n]]], {n, 1, nmax-1}];
a = Table[First@Position[tab[[j]], tabmaxs[[j]]], {j, 1, nmax-1}] // Flatten

Crossrefs

Cf. A010051, A002110, A337802, A299111, A338132.

Sequence in context: A031217 A064131 A368859 * A355077 A111497 A220554

Adjacent sequences: A338235 A338236 A338237 * A338239 A338240 A338241

Keyword

nonn

Author

Andres Cicuttin, Oct 17 2020

Status

approved