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%I #19 Jan 05 2020 06:13:08
%S 1,3,5,2,4,2,2,4,2,4,3,2,3,4,2,2,1,1,2,1,5,1,7,1,5,4,2,2,3,3,2,11,5,
%T 10,4,2,2,6,14,4,6,2,3,9,14,10,3,3,4,2,1,5,4,16,8,9,5,8,14,6,2,2,26,8,
%U 30,4,5,1,4,2,22,36,20,2,10,2,15,3,18,6,15
%N 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 semiprime (A001358).
%C Previous name was: Least integers so ascending descending base exponent transforms all semiprime.
%C Semiprime analogy to A113320. The sequence is probably infinite, but it is hard to characterize the asymptotic cost of adding an n-th term. The ascending descending base exponent transform of semiprimes is A113173.
%C The sequence is infinite because a(n) is the minimum k such that a(1)^k + k^a(1) + Sum_{i=2..n-1} a(i)^a(n-i+1) is semiprime, and since a(1)=1 this is equal to 1+k+T where T does not depend on k, thus k is the smallest positive value that makes 1+k+T semiprime, which exists because semiprimes are infinite. - _Giovanni Resta_, Jan 03 2020
%F 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 in A001358}.
%e a(1) = 1 by definition.
%e a(2) = 3 because 3 is the min x such that 1^x + x^1 is semiprime, i.e., 1^3 + 3^1 = 4 = 2*2.
%e a(3) = 5 because 1^5 + 3^3 + 5^1 = 33 = 3 * 11 is semiprime.
%e a(4) = 2 because 1^2 + 3^5 + 5^3 + 2^1 = 371 = 7 * 53.
%e a(5) = 4 because 1^4 + 3^2 + 5^5 + 2^3 + 4^1 = 3147 = 3 * 1049.
%e a(6) = 2 because 1^2 + 3^4 + 5^2 + 2^5 + 4^3 + 2^1 = 205 = 5 * 41.
%e a(7) = 2 because 1^2 + 3^2 + 5^4 + 2^2 + 4^5 + 2^3 + 2^1 = 1673 = 7 * 239.
%e a(8) = 4 because 1^4 + 3^2 + 5^2 + 2^4 + 4^2 + 2^5 + 2^3 + 4^1 = 111 = 3 * 37.
%t semipQ[n_] := PrimeOmega[n] == 2; inve[w_] := Total[w^Reverse[w]]; a[1] = 1; a[n_] := a[n] = Block[{k = 0}, While[! semipQ[ inve@ Append[ Array[a, n - 1], ++k]]]; k]; Array[a, 81] (* _Giovanni Resta_, Jun 13 2016 *)
%o (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(2!=bigomega(t+1+k), k++); a[n]=k; print1(k, ", "))} \\ _Andrew Howroyd_, Jan 03 2020
%Y Cf. A001358, A005408, A113122, A113153, A113154, A113336, A113320, A113271, A113258, A113257, A113231, A087316, A113208.
%K nonn
%O 1,2
%A _Jonathan Vos Post_, Jan 08 2006
%E Corrected and extended by _Giovanni Resta_, Jun 13 2016
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