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

%I A113257 #12 Jun 07 2025 13:02:08
%S A113257 1,5,266,268722,4682453347,2978988815561863,722638800922610642480852,
%T A113257 22529984108212742763058965679103268,
%U A113257 57286470055793196612331429228839529219232484069
%N A113257 Ascending descending base exponent transform of squares (A000290).
%C A113257 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?
%H A113257 G. C. Greubel, <a href="/A113257/b113257.txt">Table of n, a(n) for n = 1..30</a>
%F A113257 a(n) = Sum_{i=1..n} (i^2)^((n-i+1)^2).
%F A113257 a(n) = Sum_{i=1..n} (A000290(i))^(A000290(n-i+1)).
%F A113257 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
%e A113257 a(1) = 1 because (1^2)^(1^2) = 1^1 = 1.
%e A113257 a(2) = 5 because (1^2)^(4^1) + (4^1)^(1^4) = 1^4 + 4^1 = 5.
%e A113257 a(3) = 266 = 1^9 + 4^4 + 9^1.
%e A113257 a(4) = 268722 = 1^16 + 4^9 + 9^4 + 16^1.
%e A113257 a(5) = 4682453347 = 1^25 + 4^16 + 9^9 + 16^4 + 25^1.
%e A113257 a(6) = 2978988815561863 = 1^36 + 4^25 + 9^16 + 16^9 + 25^4 + 36^1.
%e A113257 a(7) = 722638800922610642480852 = 1^49 + 4^36 + 9^25 + 16^16 + 25^9 + 36^4 + 49^1.
%e A113257 a(8) = 22529984108212742763058965679103268 = 1^64 + 4^49 + 9^36 + 16^25 + 25^16 + 36^9 + 49^4 + 64^1.
%e A113257 a(9) = 57286470055793196612331429228839529219232484069 = 1^81 + 4^64 + 9^49 + 16^36 + 25^25 + 36^16 + 49^9 + 64^4 + 81^1.
%t A113257 Table[Sum[(k^2)^((n - k + 1)^2), {k, 1, n}], {n, 1, 10}] (* _G. C. Greubel_, May 18 2017 *)
%o A113257 (PARI) for(n=1,10, print1(sum(k=1,n, (k^2)^((n-k+1)^2) ), ", ")) \\ _G. C. Greubel_, May 18 2017
%Y A113257 Cf. A000290, A005408, A113122, A113153, A113154.
%K A113257 easy,nonn
%O A113257 1,2
%A A113257 _Jonathan Vos Post_, Jan 07 2006
%E A113257 a(4) and a(5) corrected by _Giovanni Resta_, Jun 13 2016