A113257 Ascending descending base exponent transform of squares (A000290).

1, 5, 266, 268722, 4682453347, 2978988815561863, 722638800922610642480852, 22529984108212742763058965679103268, 57286470055793196612331429228839529219232484069

Offset: 1

Author

Jonathan Vos Post, Jan 07 2006

Keywords

• easy
• nonn

Comments

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?

Examples

			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) = 268722 = 1^16 + 4^9 + 9^4 + 16^1.
a(5) = 4682453347 = 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.
		

Links

• G. C. Greubel, Table of n, a(n) for n = 1..30

Crossrefs

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

Programs

Mathematica

Table[Sum[(k^2)^((n - k + 1)^2), {k, 1, n}], {n, 1, 10}] (* G. C. Greubel, May 18 2017 *)

PARI

for(n=1,10, print1(sum(k=1,n, (k^2)^((n-k+1)^2) ), ", ")) \\ G. C. Greubel, May 18 2017

Formula

a(n) = Sum_{i=1..n} (i^2)^((n-i+1)^2).

a(n) = Sum_{i=1..n} (A000290(i))^(A000290(n-i+1)).

log(a(n)) ~ n^2 * (-1 + 2*LambertW(exp(1/2)*n/2))^3 / (4*LambertW(exp(1/2)*n/2)^2). - Vaclav Kotesovec, Jun 07 2025

Extensions

a(4) and a(5) corrected by Giovanni Resta, Jun 13 2016