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A001146 2^(2^n).
(Formerly M1297 N0497)
42
2, 4, 16, 256, 65536, 4294967296, 18446744073709551616, 340282366920938463463374607431768211456, 115792089237316195423570985008687907853269984665640564039457584007913129639936 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,1

COMMENTS

Or, write previous term in base 2, read in base 4.

a(1) = 2, a(n) = smallest power of 2 which does not divide the product of all previous terms.

Number of truth tables generated by boolean expressions of n variables. - C. Bradford Barber (bradb(AT)shore.net), Dec 27 2005

Comments from Ross Drewe, Feb 13 2008: (Start) Or, number of distinct n-ary operators in a binary logic. The total number of n-ary operators in a k-valued logic is T = k^(k^n), i.e. if S is a set of k elements, there are T ways of mapping an ordered subset of n elements from S to an element of S. Some operators are "degenerate": the operator has arity p, if only p of the n input values influence the output. Therefore the set of operators can be partitioned into n+1 disjoint subsets representing arities from 0 to n.

n = 2, k = 2 gives the familiar Boolean operators or functions, C = F(A,B). There are 2^2^2 = 16 operators, composed of: arity 0: 2 operators (C = 0 or 1), arity 1: 4 operators (C = A, B, not(A), not(B)), arity 2: 10 operators (including well-known pairs AND/NAND, OR/NOR, XOR/EQ). (End)

Comment from José María Grau Ribas, Jan 19 2012: Or, numbers that can be formed using the number 2, the power operator (^), and parenthesis. [The paper by Guy and Selfridge (see A003018) shows that this is the same as A001146. - N. J. A. Sloane, Jan 21 2012]

a(n) is the highest value k such that A173419(k) = n+1. - Charles R Greathouse IV, Oct 03 2012

REFERENCES

J. H. Conway, Sphere packings, lattices, codes and greed, pp. 45-55 of Proc. Intern. Congr. Math., Vol. 2, 1994.

D. E. Knuth, The Art of Computer Programming, Vol. 4A, Section 7.1.1, p. 79.

R. Ondrejka, Exact values of 2^n, n=1(1)4000, Math. Comp., 23 (1969), 456.

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..12

A. V. Aho and N. J. A. Sloane, Some doubly exponential sequences, Fib. Quart., 11 (1973), 429-437.

Jose María Grau and A. M. Oller-Marcen On the last digit and the last non-zero digit of n^n in base b.

Eric Weisstein's World of Mathematics, Irrationality Sequence, Quadratic Recurrence Equation, Coin Tossing.

Index entries for sequences of form a(n+1)=a(n)^2 + ...

FORMULA

a(n+1) = (a(n))^2

1 = sum(n>=0, a(n)/A051179(n+1) ) = 2/3 + 4/15 + 16/255 + 256/65535...; with partial sums: 2/3, 14/15, 254/255, 65534/65535... - Gary W. Adamson, Jun 15 2003

Generating function: f(x)=1/(1-2*x). Note: the generating function is not for a(n) but for log_2(a(n)). - Hieronymus Fischer, Jan 19 2006

MATHEMATICA

lst={}; Do[AppendTo[lst, 2^(2^n)], {n, 12}]; lst [From Vladimir Joseph Stephan Orlovsky, Mar 01 2009]

2^2^Range[0, 10] (* Harvey P. Dale, Jul 20 2011 *)

PROG

(MAGMA) [2^(2^n): n in [0..8]]; // Vincenzo Librandi, Jun 20 2011

(PARI) a(n)=1<<2^n \\ Charles R Greathouse IV, Jul 25 2011

(PARI) a(n)=2^2^n \\ Charles R Greathouse IV, Oct 03 2012

(Haskell)

a001146 = (2 ^) . (2 ^)

a001146_list = iterate (^ 2) 2  -- Reinhard Zumkeller, Jun 04 2012

CROSSREFS

Cf. A026477, A062090, A062091, A000215, A112535, A155538.

Sequence in context: A178077 A218148 A112535 * A114641 A152690 A194457

Adjacent sequences:  A001143 A001144 A001145 * A001147 A001148 A001149

KEYWORD

nonn,easy,nice

AUTHOR

N. J. A. Sloane.

STATUS

approved

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Last modified August 22 15:28 EDT 2014. Contains 245954 sequences.