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a(n) is the sum of the terms of the symmetric square array defined by M(i,j) = prime(i)+i-j for i >= j and M(i,j) = M(j,i) if i < j.
1

%I #43 Aug 14 2019 02:44:21

%S 2,13,44,105,224,397,660,1001,1464,2105,2866,3849,5030,6373,7946,9829,

%T 12048,14489,17310,20459,23872,27731,31972,36707,42060,47861,54022,

%U 60663,67688,75225,83902,93147,103108,113543,125014,136995,149788,163419,177760,192987,209126,225871,243912,262595,282108

%N a(n) is the sum of the terms of the symmetric square array defined by M(i,j) = prime(i)+i-j for i >= j and M(i,j) = M(j,i) if i < j.

%F a(n) = a(n-1) + (2n-1)*prime(n) + n*(n-1). - _Charlie Neder_, Jun 21 2019

%e For n=1, the array is 2, and the sum is 2.

%e .

%e . 2 4

%e For n=2, the array is and the sum is 13.

%e . 4 3

%e .

%e . 2 4 7

%e For n=3, the array is 4 3 6 and the sum is 44.

%e 7 6 5

%o (Excel, VBA)

%o Sub A308731()

%o n = 50

%o Cells(1, 1) = 2

%o p = 0

%o For i = 2 To n^2

%o isPrime = True

%o For j = 1 To p - 1

%o If i Mod Cells(j, j) = 0 Then

%o isPrime = False

%o Exit For

%o End If

%o Next j

%o If isPrime then

%o p = p + 1

%o Cells(p, p) = i

%o If p >= n Then

%o Exit For

%o End If

%o End If

%o Next i

%o For i = 2 To p

%o For j = 1 To i - 1

%o Cells(i, j) = Cells(i, i) + i - j

%o Cells(j, i) = Cells(i, j)

%o Next j

%o Next i

%o For i = 1 To n

%o Sum = 0

%o For k = 1 To i

%o For j = 1 To i

%o Sum = Sum + Cells(k, j)

%o Cells(i, n + 1) = Sum

%o Next j

%o Next k

%o Next i

%o End Sub

%o (PARI) M(i,j) = if (i>=j, prime(i)+i-j, M(j,i));

%o a(n) = sum(i=1, n, vecsum(vector(n, k, M(i,k)))); \\ _Michel Marcus_, Jun 21 2019

%Y Cf. A000040.

%K nonn

%O 1,1

%A _Ali Sada_, Jun 20 2019

%E Edited by _Michel Marcus_, Jun 21 2019