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A001146
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2^(2^n).
(Formerly M1297 N0497)
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38
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2, 4, 16, 256, 65536, 4294967296, 18446744073709551616, 340282366920938463463374607431768211456, 115792089237316195423570985008687907853269984665640564039457584007913129639936
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,1
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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 (rd(AT)labyrinth.net.au), 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 (grau.ribas(AT)gmail.com), 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]
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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).
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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.
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 + ...
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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 (qntmpkt(AT)yahoo.com), Jun 15 2003
Generating function: f(x)=1/(1-2*x). Note: the generating function is not for a(n) but for for log_2(a(n)). - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Jan 19 2006
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MATHEMATICA
| lst={}; Do[AppendTo[lst, 2^(2^n)], {n, 12}]; lst [From Vladimir Orlovsky (4vladimir(AT)gmail.com), Mar 01 2009]
2^2^Range[0, 10] (* From Harvey P. Dale, Jul 20 2011 *)
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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
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CROSSREFS
| Cf. A026477, A062090, A062091, A000215, A112535, A155538.
Sequence in context: A105788 A071008 A178077 * A114641 A152690 A194457
Adjacent sequences: A001143 A001144 A001145 * A001147 A001148 A001149
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KEYWORD
| nonn,easy,nice
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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