1,3

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.

a(1) = 1. For n>1: a(n) = min {n>0: SUM[from i = 1 to n] (a(i))^(a(n-i+1)) is prime}.

a(1) = 1 by definition.

a(2) = 1 because 1 is the min such that 1^a(2) + a(2)^1 is prime (p=2).

a(3) = 3 because 3 is the min such that 1^a(3) + 1^1 + a(3)^1 is prime (p=5).

a(4) = 2 because 2 is the min such that 1^a(4) + 1^1 + 3^1 + a(4)^1 is prime (p=7).

a(5) = 6 because 6 is the min such that 1^a(5) + 1^2 + 3^3 + 2^1 + a(5)^1 is prime (p=37).

a(6) = 4 = min such that 1^a(6) + 1^6 + 3^2 + 2^3 + 6^1 + a(6)^1 is prime (p=29).

a(7) = 12 because 1^12 + 1^4 + 3^6 + 2^2 + 6^3 + 4^1 + 12^1 = 967 is prime.

a(8) = 4 because 1^4 + 1^12 + 3^4 + 2^6 + 6^2 + 4^3 + 12^1 + 4^1 = 263 is prime.

a(9) = 6 because 1^6 + 1^4 + 3^12 + 2^4 + 6^6 + 4^2 + 12^3 + 4^1 + 6^1 = 579869 is prime.

a(10) = 2 because 1^2 + 1^6 + 3^4 + 2^12 + 6^4 + 4^6 + 12^2 + 4^3 + 6^1 + 2^1 = 9787 is prime.

a(11) = 12 because 1^12 + 1^2 + 3^6 + 2^4 + 6^12 + 4^4 + 12^6 + 4^2 + 6^3 + 2^1 + 12^1 = 2179769569 is prime.

a(12) = 28 because 1^28 + 1^12 + 3^2 + 2^6 + 6^4 + 4^12 + 12^4 + 4^6 + 6^2 + 2^3 + 12^1 + 28^1 = 16803503 is prime.

a(13) = 16 because 1^16 + 1^28 + 3^12 + 2^2 + 6^6 + 4^4 + 12^12 + 4^4 + 6^6 + 2^2 + 12^3 + 28^1 + 16^1 = 8916101075303 is prime.

Cf. A000040, A005408, A113122, A113153, A113154.

Adjacent sequences: A113317 A113318 A113319 this_sequence A113321 A113322 A113323

Sequence in context: A105354 A094077 A091018 this_sequence A092401 A116626 A074323

easy,nonn

Jonathan Vos Post (jvospost2(AT)yahoo.com), Jan 07 2006