A001146 a(n) = 2^(2^n). (Formerly M1297 N0497)

2, 4, 16, 256, 65536, 4294967296, 18446744073709551616, 340282366920938463463374607431768211456, 115792089237316195423570985008687907853269984665640564039457584007913129639936

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

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.

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 well-known pairs AND/NAND, OR/NOR, XOR/EQ). (End)

From José María Grau Ribas, Jan 19 2012: (Start)

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]

a(n) is the highest value k such that A173419(k) = n+1. - Charles R Greathouse IV, Oct 03 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^n-1). 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^4-1 divides 2^n-1, 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

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

A. V. Aho and N. J. A. Sloane, Some doubly exponential sequences, Fibonacci Quarterly, Vol. 11, No. 4 (1973), pp. 429-437, alternative link.

Jan Brandts and Apo Cihangir, Enumeration and investigation of acute 0/1-simplices modulo the action of the hyperoctahedral group, Special Matrices, Vol. 5, No. 1 (2017), pp. 158-201, arXiv preprint, arXiv:1512.03044 [math.CO], 2015.

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

Jose María Grau and A. M. Oller-Marcén On the last digit and the last non-zero digit of n^n in base b, arXiv:1203.4066 [math.NT], 2012.

Richard K. Guy and J. L. Selfridge, The nesting and roosting habits of the laddered parenthesis, Amer. Math. Monthly 80 (8) (1973), 868-876.

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

Rudolf Ondrejka, Letter to N. J. A. Sloane, May 15 1976

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

a(n) = A000079(A000079(n)). - Robert Israel, Jan 15 2015

Sum_{n>=0} 1/a(n) = A007404. - Amiram Eldar, Oct 14 2020

From Amiram Eldar, Jan 28 2021: (Start)

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

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

Maple

A001146:=n->2^(2^n): seq(A001146(n), n=0..9); # Wesley Ivan Hurt, Sep 19 2014

Mathematica

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
(Python)
def A001146(n): return 1<<(1<<n) # Chai Wah Wu, Mar 14 2023

Crossrefs

Cf. A000079, A000215, A007404, A026477, A062090, A062091, A112535, A155538, A215016.

Cf. also A003018, A051179, A173419, A165420, A247165, A247219.

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