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A113320 a(1)=1 and a(n) for n>1 has the smallest positive value such that Sum_{i=1..n} a(i)^a(n-i+1) is prime.

%I A113320 #20 Jan 05 2020 06:07:15
%S A113320 1,1,1,2,2,4,4,4,6,2,6,4,18,6,4,20,6,30,4,40,30,8,18,16,40,128,24,40,
%T A113320 58,194,78,84,56,56,72,112,98,300,444,54,978,1938,120,126,6,1750
%N A113320 a(1)=1 and a(n) for n>1 has the smallest positive value such that Sum_{i=1..n} a(i)^a(n-i+1) is prime.
%C A113320 Previous name was: Least integers so ascending descending base exponent transforms all prime.
%C A113320 This is the first sequence submitted as a solution to an "ascending descending base exponent transform inverse problem" where the sequence is iteratively defined such that the transform meets a constraint. The sequence is infinite, but it is hard to characterize the asymptotic cost of adding an n-th term. A003101 is the ascending descending base exponent transform of natural numbers A000027. The ascending descending base exponent transform applied to the Fibonacci numbers is A113122; applied to the tribonacci numbers is A113153; applied to the Lucas numbers is A113154.
%F A113320 a(1) = 1. For n>1, a(n) = min {k>0: a(1)^k + k^a(1) + Sum_{i=2..n-1} a(i)^a(n-i+1) is prime}.
%e A113320 a(1) = 1 by definition.
%e A113320 a(2) = 1 because 1 is the min such that 1^a(2) + a(2)^1 is prime (p=2).
%e A113320 a(3) = 1 because 1 is the min such that 1^a(3) + 1^1 + a(3)^1 is prime (p=5).
%e A113320 a(4) = 2 because 2 is the min such that 1^a(4) + 1^1 + 3^1 + a(4)^1 is prime (p=7).
%t A113320 inve[w_] := Total[w^Reverse[w]]; a[1] = 1; a[n_] := a[n] = Block[{k = 0}, While[! PrimeQ[ inve@ Append[Array[a, n-1], ++k]]]; k]; Array[a, 46] (* _Giovanni Resta_, Jun 13 2016 *)
%o A113320 (PARI) lista(n)={my(a=vector(n)); a[1]=1; print1(1, ", "); for(n=2, #a, my(t=sum(i=2, n-1, a[i]^a[n-i+1])); my(k=1); while(!ispseudoprime(t+1+k), k++); a[n]=k; print1(k, ", "))} \\ _Andrew Howroyd_, Jan 03 2020
%Y A113320 Cf. A000040, A005408, A113122, A113153, A113154.
%K A113320 easy,nonn
%O A113320 1,4
%A A113320 _Jonathan Vos Post_, Jan 07 2006
%E A113320 Corrected and extended by _Giovanni Resta_, Jun 13 2016
%E A113320 New name from _Giovanni Resta_, Jan 03 2020