

A001146


a(n) = 2^(2^n).
(Formerly M1297 N0497)


101



2, 4, 16, 256, 65536, 4294967296, 18446744073709551616, 340282366920938463463374607431768211456, 115792089237316195423570985008687907853269984665640564039457584007913129639936
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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
Or, number of distinct nary operators in a binary logic. The total number of nary operators in a kvalued 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.
For 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 wellknown pairs AND/NAND, OR/NOR, XOR/EQ). (End)
Or, numbers that can be formed using the number 2, the power operator (^), and parenthesis. (End) [The paper by Guy and Selfridge (see also A003018) shows that this is the same as the current sequence.  N. J. A. Sloane, Jan 21 2012]
Let b(0) = 8 and b(n+1) = the smallest number not in the sequence such that b(n+1)  Product_{i=0..n} b(i) divides b(n+1)*Product_{i=0..n} b(i). Then b(n) = a(n) for n > 0.  Derek Orr, Jan 15 2015
Twice the number of distinct minimal toss sequences of a coin to obtain all sequences of length n, which is 2^(2^n1). This derives from the 2^n ways to cut each of the De Bruijn sequences B(2,n).  Maurizio De Leo, Feb 28 2015
I conjecture that { a(n) ; n>1 } are the numbers such that n^41 divides 2^n1, intersection of A247219 and A247165.  M. F. Hasler, Jul 25 2015


REFERENCES

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



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
Product_{n>=0} (1 + 1/a(n)) = 2.
Product_{n>=0} (1  1/a(n)) = A215016. (End)


MAPLE



MATHEMATICA



PROG

(Haskell)
a001146 = (2 ^) . (2 ^)
(Python)


CROSSREFS



KEYWORD

nonn,easy,nice


AUTHOR



STATUS

approved



