0,2

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. The smallest primes in this (always odd) sequence are a(1) = 3, a(3) = 41 and a(5) = 543. What is the next prime?

a(n) = SUM[from i = 0 to n] (2^i)^(2^(n-i+1)). a(n) = SUM[from i = 0 to n] (2^(n-i+1))^(2^i). a(n) = SUM[from i = 0 to n] (A000079(i))^(A000079(n-i+1)).

a(0) = 1 because (2^0)^(2^0) = 1^1 = 1.

a(1) = 3 = (2^0)^(2^1) + (2^1)^(2^0) = 1^2 + 2^1.

a(2) = 9 = (2^0)^(2^2) + (2^1)^(2^1) + (2^2)^(2^0) = 1^4 + 2^2 + 4^1.

a(3) = 41 = 1^8 + 2^4 + 4^2 + 8^1.

a(4) = 543 = 1^16 + 2^8 + 4^4 + 8^2 + 16^1

a(5) = 135457 = 1^32 + 2^16 + 4^8 + 8^4 + 16^2 + 32^1.

a(6) = 8606778433 = 1^64 + 2^32 + 4^16 + 8^8 + 16^4 + 32^2 + 64^1.

a(7) = 36893769626691833985 = 1^128 + 2^64 + 4^32 + 8^16 + 16^8 + 32^4 + 64^2 + 128^1.

a(8) = 680564733921105089459460297630318346497 = 1^256 + 2^128 + 4^64 + 8^32 + 16^16 + 32^8 + 64^4 + 128^2 + 256^1.

Cf. A000079, A005408, A113122, A113153, A113154.

Adjacent sequences: A113268 A113269 A113270 this_sequence A113272 A113273 A113274

Sequence in context: A074502 A109743 A139150 this_sequence A057724 A129879 A001427

easy,nonn

Jonathan Vos Post (jvospost3(AT)gmail.com), Jan 07 2006