Sums of Sums Sequences

MAIN We may define sequences in the following format, a(1) = ^2 f(i_1) \\ a(2) = ^2 ^{f(i_1)} f(i_2) \\ a(3) = ^2 ^{f(i_1)} ^{f(i_2)} f(i_3) \\ \cdots \\ a(n) = ^2 ^{f(i_1)} ^{f(i_2)}\cdots^{f(i_{n-1})} f(i_n) for some f. This can also be extended with some g a(n) = ^2 ^{f(i_1)} ^{f(i_2)}\cdots^{f(i_{n-1})} f(i_n) for example, it would appear that for f(x)=x + 2 and g(n)=n, we have a(n) = ^2 ^{f(i_1)} ^{f(i_2)}\cdots^{f(i_{n-1})} f(i_n) = {n+2}{n+3} and for for f(x)=x + 1 and g(n)=n a(n) = ^2 ^{f(i_1)} ^{f(i_2)}\cdots^{f(i_{n-1})} f(i_n) = (n+1)^2+1 To keep this as general as possible define the following \Sigma(a,b,c,d;n)= ^{a(n)} ^{c(i_1)} ^{c(i_2)}\cdots^{c(i_{n-1})} d(i_n) Then for example we have \Sigma(1,1,n+1,n;n)= {(n+2)!(n-1)!} which is A000245. LIST \Sigma(1,1,1,1;n)=1 \\ \Sigma(n,n,n,n;n)=n \\ \Sigma(n,n,n,f(n);n)=f(n) \\ \Sigma(n,n-1,n,n;n)=n^2 \\ \Sigma(n,n-2,n,n;n)={2} \\ \Sigma(1,n,n+1,n;n)=n \\ \Sigma(1,1,n,n;n)=1 \\ \Sigma(1,1,n+1,n;n)= {(n+2)!(n-1)!} \\ \Sigma(n+1,n+1,n+1,n+1;n) = 3,13,55,231,966,\cdots I have found a very special conjecture \Sigma(1,1,\pi(n+1)+1,n;n) = 3(2(n-1))+0^{n-1} which is true for the first 17 terms. Where π(x) is the prime counting function. ^{a(n)} ^{c(i_1)} ^{c(i_2)}\cdots^{c(i_{n-1})} i_n = 3(2(n-1))+0^{n-1} this most likely is due to the sub linear growth of pi(n). INTEGRALS We also may write a series of integrals I_n = ^{a(n)} ^{c(i_1)} ^{c(i_2)}\cdots^{c(i_{n-1})} d(i_n)\;di_n\;\cdots di_2 di_1 If we let. a(n)=1, b(n)=0, c(n)=n + 1, d(n)=n + 1, This appears to give A000272/(n + 1)!. If we let. a(n)=1, b(n)=0, c(n)=n + 1, d(n)=n, This appears to give A213326/(n + 1)!. If we let. a(n)=n, b(n)=0, c(n)=n + 1, d(n)=1, This appears to give A193678/n! If we let. a(n)=2n, b(n)=0, c(n)=n, d(n)=1, This appears to give A062971/n! If we let. a(n)=n, b(n)=0, c(n)=n, d(n)=1, This appears to give nn/n!