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A308731 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.

%I A308731 #47 May 09 2025 00:59:34
%S A308731 2,13,44,105,224,397,660,1001,1464,2105,2866,3849,5030,6373,7946,9829,
%T A308731 12048,14489,17310,20459,23872,27731,31972,36707,42060,47861,54022,
%U A308731 60663,67688,75225,83902,93147,103108,113543,125014,136995,149788,163419,177760,192987,209126,225871,243912,262595,282108
%N A308731 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 A308731 a(n) = a(n-1) + (2n-1)*prime(n) + n*(n-1). - _Charlie Neder_, Jun 21 2019
%F A308731 a(n) = A316322(n) + A007290(n+1). - _M. F. Hasler_, May 08 2025
%e A308731 For n=1, the array is 2, and the sum is 2.
%e A308731 .
%e A308731 .                     2  4
%e A308731 For n=2, the array is      and the sum is 13.
%e A308731 .                     4  3
%e A308731 .
%e A308731 .                     2  4  7
%e A308731 For n=3, the array is 4  3  6 and the sum is 44.
%e A308731                       7  6  5
%o A308731 (Excel, VBA)
%o A308731 Sub A308731()
%o A308731 n = 50
%o A308731 Cells(1, 1) = 2
%o A308731 p = 0
%o A308731 For i = 2 To n^2
%o A308731     isPrime = True
%o A308731     For j = 1 To p - 1
%o A308731         If i Mod Cells(j, j) = 0 Then
%o A308731             isPrime = False
%o A308731             Exit For
%o A308731         End If
%o A308731     Next j
%o A308731     If isPrime then
%o A308731         p = p + 1
%o A308731         Cells(p, p) = i
%o A308731         If p >= n Then
%o A308731             Exit For
%o A308731         End If
%o A308731     End If
%o A308731 Next i
%o A308731 For i = 2 To p
%o A308731     For j = 1 To i - 1
%o A308731         Cells(i, j) = Cells(i, i) + i - j
%o A308731         Cells(j, i) = Cells(i, j)
%o A308731     Next j
%o A308731 Next i
%o A308731 For i = 1 To n
%o A308731     Sum = 0
%o A308731     For k = 1 To i
%o A308731         For j = 1 To i
%o A308731             Sum = Sum + Cells(k, j)
%o A308731             Cells(i, n + 1) = Sum
%o A308731         Next j
%o A308731     Next k
%o A308731 Next i
%o A308731 End Sub
%o A308731 (PARI) M(i,j) = if (i>=j, prime(i)+i-j, M(j,i));
%o A308731 a(n) = sum(i=1, n, vecsum(vector(n, k, M(i,k)))); \\ _Michel Marcus_, Jun 21 2019
%o A308731 (PARI) A308731_first(N)=vector(N, n, N+=if(n>1, prime(n)*(2*n-1)+n*(n-1), 2-N)) \\ This is a more efficient way to compute the list [a(1), ..., a(N)]
%o A308731 apply( {A308731(n)=sum(k=1,n,prime(k)*(2*k-1))+2*binomial(n+1,3)}, [1..20]) \\ _M. F. Hasler_, May 08 2025
%Y A308731 Cf. A000040, A007290 (partial sums of n(n-1)), A316322 (sum of "pile of primes").
%K A308731 nonn
%O A308731 1,1
%A A308731 _Ali Sada_, Jun 20 2019
%E A308731 Edited by _Michel Marcus_, Jun 21 2019