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A113271 Ascending descending base exponent transform of 2^n.

%I A113271 #13 Jun 07 2025 05:52:29
%S A113271 1,3,9,41,593,135457,8606778433,36893769626691833985,
%T A113271 680564733921105089459460297630318346497,
%U A113271 231584178474632390853419071752762496470716041121409734167406717963826481922561
%N A113271 Ascending descending base exponent transform of 2^n.
%C A113271 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?
%H A113271 G. C. Greubel, <a href="/A113271/b113271.txt">Table of n, a(n) for n = 0..12</a>
%F A113271 a(n) = Sum_{i=0..n} (2^i)^(2^(n-i)).
%F A113271 a(n) = Sum_{i=0..n} (2^(n-i))^(2^i).
%F A113271 a(n) = Sum_{i=0..n} (A000079(i))^(A000079(n-i)).
%F A113271 a(n) ~ 2^(2^(n-1) + 1). - _Vaclav Kotesovec_, Jun 07 2025
%e A113271 a(0) = 1 because (2^0)^(2^0) = 1^1 = 1.
%e A113271 a(1) = 3 = (2^0)^(2^1) + (2^1)^(2^0) = 1^2 + 2^1.
%e A113271 a(2) = 9 = (2^0)^(2^2) + (2^1)^(2^1) + (2^2)^(2^0) = 1^4 + 2^2 + 4^1.
%e A113271 a(3) = 41 = 1^8 + 2^4 + 4^2 + 8^1.
%e A113271 a(4) = 593 = 1^16 + 2^8 + 4^4 + 8^2 + 16^1
%e A113271 a(5) = 135457 = 1^32 + 2^16 + 4^8 + 8^4 + 16^2 + 32^1.
%e A113271 a(6) = 8606778433 = 1^64 + 2^32 + 4^16 + 8^8 + 16^4 + 32^2 + 64^1.
%e A113271 a(7) = 36893769626691833985 = 1^128 + 2^64 + 4^32 + 8^16 + 16^8 + 32^4 + 64^2 + 128^1.
%e A113271 a(8) = 680564733921105089459460297630318346497 = 1^256 + 2^128 + 4^64 + 8^32 + 16^16 + 32^8 + 64^4 + 128^2 + 256^1.
%t A113271 Table[Sum[(2^k)^(2^(n - k)), {k, 0, n}], {n, 0, 10}] (* _G. C. Greubel_, May 19 2017 *)
%o A113271 (PARI) for(n=0,5, print1(sum(k=0,n, (2^k)^(2^(n-k))), ", ")) \\ _G. C. Greubel_, May 19 2017
%Y A113271 Cf. A000079, A005408, A113122, A113153, A113154.
%K A113271 easy,nonn
%O A113271 0,2
%A A113271 _Jonathan Vos Post_, Jan 07 2006
%E A113271 a(4) corrected by _Giovanni Resta_, Jun 13 2016
%E A113271 Formulas corrected by _G. C. Greubel_, May 19 2017