A001146 a(n) = 2^(2^n).

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

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy
• nice

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

Erdős has shown that it is an irrationality sequence (see Guy reference). - Stefano Spezia, Oct 13 2024

References

• R. K. Guy, Unsolved Problems in Number Theory, Springer, 1st edition, 1981. See section E24.
• 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..11
• A. V. Aho and N. J. A. Sloane, Some doubly exponential sequences, Fibonacci Quarterly, Vol. 11, No. 4 (1973), pp. 429-437.
• A. V. Aho and N. J. A. Sloane, Some doubly exponential sequences, Fibonacci Quarterly, Vol. 11, No. 4 (1973), pp. 429-437 (original plus references that F.Q. forgot to include - see last page!)
• 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 + ...

Crossrefs

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

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

Programs

Haskell

a001146 = (2 ^) . (2 ^)
a001146_list = iterate (^ 2) 2  -- Reinhard Zumkeller, Jun 04 2012

Magma

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

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 *)

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

Python

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

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)