A299111 Maximum value of the cyclic convolution of first n primes with themselves.

4, 13, 37, 82, 183, 344, 601, 918, 1355, 2048, 2873, 3978, 5455, 7112, 9105, 11530, 14391, 17504, 21353, 25686, 30311, 35536, 41421, 48010, 55911, 64632, 73869, 83766, 94151, 105420, 118569, 132566, 148247, 164564, 182617, 201770, 222975, 245532, 269253

Offset: 1

Author

Andres Cicuttin, Feb 02 2018

Keywords

• nonn

Examples

			For n = 4 the four possible cyclic convolution of first four primes with themselves are:
(2,3,5,7).(7,5,3,2) = 2*7 + 3*5 + 5*3 + 7*2 = 14 + 15 + 15 + 14 = 58,
(2,3,5,7).(2,7,5,3) = 2*2 + 3*7 + 5*5 + 7*3 = 4 + 21 + 25 + 21 = 71,
(2,3,5,7).(3,2,7,5) = 2*3 + 3*2 + 5*7 + 7*5 = 6 + 6 + 35 + 35 = 82,
(2,3,5,7).(5,3,2,7) = 2*5 + 3*3 + 5*2 + 7*7 = 10 + 9 + 10 + 49 = 78,
then a(4)=82 because 82 is the maximum among the four values.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..2000

Crossrefs

Cf. A014342, A299053.

Programs

Maple

f:= proc(n) local V,R,i;
    V:= Vector(n, ithprime);
    R:= ArrayTools:-FlipDimension(V,1)^%T;
    max(seq(ArrayTools:-CircularShift(R,i) . V, i=0..n-1))
end proc:
map(f, [$1..100]); # Robert Israel, Feb 07 2018

Mathematica

a[n_]:=Prime[Range[n]];
Table[Max@Table[a[n].RotateRight[Reverse[a[n]], j], {j, 0, n - 1}], {n,1,36}]

PARI

a(n) = my(vp=primes(n)); vecmax(vector(n, k, sum(i=1, n, vp[n-i+1]*vp[1+(i+k)%n]))); \\ Michel Marcus, Feb 07 2018; Jun 15 2022

Formula

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

a(n) >= A014342(n). Does the ratio a(n)/A014342(n) have a limit as n -> infinity? - Robert Israel, Feb 07 2018