A299053 Minimum value of the cyclic autocorrelation of first n primes.

4, 12, 31, 62, 133, 224, 377, 558, 865, 1304, 1805, 2462, 3337, 4280, 5389, 6726, 8449, 10264, 12663, 15294, 18061, 21200, 24961, 29166, 34173, 39508, 45017, 50870, 57141, 63788, 72299, 81234, 91365, 101732, 113327, 125166, 138355, 152348, 167179, 182862

Offset

1,1

Comments

Maximum values of the cyclic autocorrelation of first n primes are in A024450.

If we use this definition with integers instead of primes it is obtained A088003.

Links

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Formula

a(n) = Min_{k=1..n} Sum_{i=1..n} prime(i)*prime(1 + (i+k) mod n).

Example

For n = 4 the four possible cyclic autocorrelations of first four primes are:
(2,3,5,7).(2,3,5,7) = 2*2 + 3*3 + 5*5 + 7*7 = 4 + 9 + 25 + 49 = 87,
(2,3,5,7).(7,2,3,5) = 2*7 + 3*2 + 5*3 + 7*5 = 14 + 6 + 15 + 35 = 70,
(2,3,5,7).(5,7,2,3) = 2*5 + 3*7 + 5*2 + 7*3 = 10 + 21 + 10 + 21 = 62,
(2,3,5,7).(3,5,7,2) = 2*3 + 3*5 + 5*7 + 7*2 = 6 + 15 + 35 + 14 = 70,
then a(4)=62 because 62 is the minimum among the four values.

Maple

a:= n-> min(seq(add(ithprime(i)*ithprime(irem(i+k, n)+1), i=1..n), k=1..n)):
seq(a(n), n=1..40);  # Alois P. Heinz, Feb 06 2018

Mathematica

p[n_]:=Prime[Range[n]];
Table[Table[p[n].RotateRight[p[n], j], {j, 0, n-1}]//Min, {n, 1, 36}]

Prog

(PARI) a(n) = vecmin(vector(n, k, sum(i=1, n, prime(i)*prime(1+(i+k)%n)))); \\ Michel Marcus, Feb 07 2018

Crossrefs

Cf. A024450, A014342, A299111, A088003.

Sequence in context: A074252 A297079 A074210 * A005289 A037255 A027658

Adjacent sequences: A299050 A299051 A299052 * A299054 A299055 A299056

Keyword

nonn

Author

Andres Cicuttin, Feb 01 2018

Status

approved