A319012 a(n) = Sum_{i=1..n} prime(n*(i - 1) + i).

2, 9, 36, 99, 224, 407, 724, 1129, 1700, 2451, 3382, 4543, 5986, 7661, 9724, 12041, 14762, 17891, 21482, 25499, 29998, 35083, 40644, 46873, 53620, 61077, 69240, 78119, 87686, 98053, 109290, 121503, 134388, 148297, 162970, 178905, 195770, 213725, 232794

Offset

1,1

Comments

a(n) is the trace of the n X n matrix M(n) whose first row contains the first n primes in increasing order, the second row of M(n) contains the next n primes in increasing order, and so on (see examples below).

Conjecture: a(2) and a(3) are the only terms that are perfect squares.

Links

Stefano Spezia, Table of n, a(n) for n = 1..3000

Formula

a(n) = Sum_{i=1..n} A000040(n*(i - 1) + i).

a(n) = Sum_{k=1..n} A000040(A209297(n, k)). - Michel Marcus, Mar 18 2020

a(n) ~ n^3*log(n). - Stefano Spezia, Jul 01 2021

Example

For n = 1 the matrix M(1) is
   2
with trace Tr(M(1)) = a(1) = 2.
For n = 2 the matrix M(2) is
   2,  3
   5,  7
with Tr(M(2)) = a(2) = 9.
For n = 3 the matrix M(3) is
   2,  3, 5
   7, 11, 13
  17, 19, 23
with Tr(M(3)) = a(3) = 36.

Maple

a:=n->add(ithprime(n*(i-1)+i), i=1..n): seq(a(n), n=1..40); # Muniru A Asiru, Sep 17 2018

Mathematica

Table[Tr[Partition[Array[Prime, n^2], n]], {n, 40}]

Prog

(PARI) a(n) = sum(i=1, n, prime(n*(i - 1) + i)); \\ Michel Marcus, Sep 07 2018

Crossrefs

Cf. A000040, A067276 (determinant of the matrices M).

Cf. A209297.

Sequence in context: A175231 A175261 A086556 * A192694 A192702 A073989

Adjacent sequences: A319009 A319010 A319011 * A319013 A319014 A319015

Keyword

nonn

Author

Stefano Spezia, Sep 07 2018

Status

approved