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A113271
Ascending descending base exponent transform of 2^n.
9
1, 3, 9, 41, 593, 135457, 8606778433, 36893769626691833985, 680564733921105089459460297630318346497, 231584178474632390853419071752762496470716041121409734167406717963826481922561
OFFSET
0,2
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 primes in this (always odd) sequence are a(1) = 3, a(3) = 41 and a(5) = 543. What is the next prime?
LINKS
FORMULA
a(n) = Sum_{i=0..n} (2^i)^(2^(n-i)).
a(n) = Sum_{i=0..n} (2^(n-i))^(2^i).
a(n) = Sum_{i=0..n} (A000079(i))^(A000079(n-i)).
EXAMPLE
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) = 593 = 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.
MATHEMATICA
Table[Sum[(2^k)^(2^(n - k)), {k, 0, n}], {n, 0, 10}] (* G. C. Greubel, May 19 2017 *)
PROG
(PARI) for(n=0, 5, print1(sum(k=0, n, (2^k)^(2^(n-k))), ", ")) \\ G. C. Greubel, May 19 2017
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Jonathan Vos Post, Jan 07 2006
EXTENSIONS
a(4) corrected by Giovanni Resta, Jun 13 2016
Formulas corrected by G. C. Greubel, May 19 2017
STATUS
approved