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

%I A319012 #34 Jul 01 2021 16:06:23
%S A319012 2,9,36,99,224,407,724,1129,1700,2451,3382,4543,5986,7661,9724,12041,
%T A319012 14762,17891,21482,25499,29998,35083,40644,46873,53620,61077,69240,
%U A319012 78119,87686,98053,109290,121503,134388,148297,162970,178905,195770,213725,232794
%N A319012 a(n) = Sum_{i=1..n} prime(n*(i - 1) + i).
%C A319012 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).
%C A319012 Conjecture: a(2) and a(3) are the only terms that are perfect squares.
%H A319012 Stefano Spezia, <a href="/A319012/b319012.txt">Table of n, a(n) for n = 1..3000</a>
%F A319012 a(n) = Sum_{i=1..n} A000040(n*(i - 1) + i).
%F A319012 a(n) = Sum_{k=1..n} A000040(A209297(n, k)). - _Michel Marcus_, Mar 18 2020
%F A319012 a(n) ~ n^3*log(n). - _Stefano Spezia_, Jul 01 2021
%e A319012 For n = 1 the matrix M(1) is
%e A319012    2
%e A319012 with trace Tr(M(1)) = a(1) = 2.
%e A319012 For n = 2 the matrix M(2) is
%e A319012    2,  3
%e A319012    5,  7
%e A319012 with Tr(M(2)) = a(2) = 9.
%e A319012 For n = 3 the matrix M(3) is
%e A319012    2,  3, 5
%e A319012    7, 11, 13
%e A319012   17, 19, 23
%e A319012 with Tr(M(3)) = a(3) = 36.
%p A319012 a:=n->add(ithprime(n*(i-1)+i),i=1..n): seq(a(n),n=1..40); # _Muniru A Asiru_, Sep 17 2018
%t A319012 Table[Tr[Partition[Array[Prime, n^2], n]], {n, 40}]
%o A319012 (PARI) a(n) = sum(i=1, n, prime(n*(i - 1) + i)); \\ _Michel Marcus_, Sep 07 2018
%Y A319012 Cf. A000040, A067276 (determinant of the matrices M).
%Y A319012 Cf. A209297.
%K A319012 nonn
%O A319012 1,1
%A A319012 _Stefano Spezia_, Sep 07 2018