1,2

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 prime in this sequence is a(2) = 5. What is the next prime? What is the first square value after 1?

a(n) = SUM[from i = 1 to n] (i^2)^((n-i+1)^2). a(n) = SUM[from i = 1 to n] (A000290(i))^(A000290(n-i+1)).

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

a(2) = 5 because (1^2)^(4^1) + (4^1)^(1^4) = 1^4 + 4^1 = 5.

a(3) = 266 = 1^9 + 4^4 + 9^1.

a(4) = 266257 = 1^16 + 4^9 + 9^4 + 16^1.

a(5) = 4429250586 = 1^25 + 4^16 + 9^9 + 16^4 + 25^1.

a(6) = 2978988815561863 = 1^36 + 4^25 + 9^16 + 16^9 + 25^4 + 36^1.

a(7) = 722638800922610642480852 = 1^49 + 4^36 + 9^25 + 16^16 + 25^9 + 36^4 + 49^1.

a(8) = 22529984108212742763058965679103268 = 1^64 + 4^49 + 9^36 + 16^25 + 25^16 + 36^9 + 49^4 + 64^1.

a(9) = 57286470055793196612331429228839529219232484069 = 1^81 + 4^64 + 9^49 + 16^36 + 25^25 + 36^16 + 49^9 + 64^4 + 81^1.

Cf. A000290, A005408, A113122, A113153, A113154.

Sequence in context: A079681 A086656 A034602 this_sequence A140001 A153322 A066210

Adjacent sequences: A113254 A113255 A113256 this_sequence A113258 A113259 A113260

easy,nonn

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