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

1,1

Links

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

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

Example

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.

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}]

Prog

(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

Crossrefs

Cf. A014342, A299053.

Sequence in context: A003727 A103082 A279111 * A324250 A226866 A048474

Adjacent sequences: A299108 A299109 A299110 * A299112 A299113 A299114

Keyword

nonn

Author

Andres Cicuttin, Feb 02 2018

Status

approved