A215016 -id:A215016 - cp's OEIS frontend

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

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

Offset: 0

N. J. A. Sloane

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

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).

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 + ...

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

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

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

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

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

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

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

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

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

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)

A014571 Consider the Morse-Thue sequence (A010060) as defining a binary constant and convert it to decimal.

Original entry on oeis.org

4, 1, 2, 4, 5, 4, 0, 3, 3, 6, 4, 0, 1, 0, 7, 5, 9, 7, 7, 8, 3, 3, 6, 1, 3, 6, 8, 2, 5, 8, 4, 5, 5, 2, 8, 3, 0, 8, 9, 4, 7, 8, 3, 7, 4, 4, 5, 5, 7, 6, 9, 5, 5, 7, 5, 7, 3, 3, 7, 9, 4, 1, 5, 3, 4, 8, 7, 9, 3, 5, 9, 2, 3, 6, 5, 7, 8, 2, 5, 8, 8, 9, 6, 3, 8, 0, 4, 5, 4, 0, 4, 8, 6, 2, 1, 2, 1, 3, 3, 3, 9, 6, 2, 5, 6

Offset: 0

Eric W. Weisstein

This constant is transcendental (Mahler, 1929). - Amiram Eldar, Nov 14 2020

			0.412454033640107597783361368258455283089...
In hexadecimal, .6996966996696996... .

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 6.8 Prouhet-Thue-Morse Constant, p. 437.

Harry J. Smith, Table of n, a(n) for n = 0..20000
Boris Adamczewski and Yann Bugeaud, A short proof of the transcendence of Thue-Morse continued fractions, The American Mathematical Monthly, Vol. 114, No. 6 (2007), pp. 536-540; alternative link.
Jean-Paul Allouche and Jeffrey Shallit, The ubiquitous Prouhet-Thue-Morse sequence, in: C. Ding, T. Helleseth, and H. Niederreiter (eds.), Sequences and their applications, Springer, London, 1999, pp. 1-16; alternative link.
Joerg Arndt, Matters Computational (The Fxtbook), p.726 ff.
Michel Dekking, Transcendance du nombre de Thue-Morse, Comptes Rendus de l'Academie des Sciences de Paris, Série A, Vol. 285 (1977) A157-A160.
Arturas Dubickas, On the distance from a rational power to the nearest integer, Journal of Number Theory, Volume 117, Issue 1, March 2006, Pages 222-239.
Kurt Mahler, Arithmetische Eigenschaften der Lösungen einer Klasse von Funktionalgleichungen, Mathematische Annalen, Vol. 101 (1929), pp. 342-366, alternative link.
R. Schroeppel and R. W. Gosper, HACKMEM #122 (1972).
Eric Weisstein's World of Mathematics, Thue-Morse Constant.
Index entries for transcendental numbers

Cf. A000069, A001969, A010060, A058631, A215016, A247950.

A014571 := proc()
    local nlim,aold,a ;
    nlim := ilog2(10^Digits) ;
    aold := add( A010060(n)/2^n,n=0..nlim) ;
    a := 0.0 ;
    while abs(a-aold) > abs(a)/10^(Digits-3) do
        aold := a;
        nlim := nlim+200 ;
        a := add( A010060(n)/2^n,n=0..nlim) ;
    od:
    evalf(%/2) ;
end:
A014571() ; # R. J. Mathar, Mar 03 2008

digits = 105; t[0] = 0; t[n_?EvenQ] := t[n] = t[n/2]; t[n_?OddQ] := t[n] = 1-t[(n-1)/2]; FromDigits[{t /@ Range[digits*Log[10]/Log[2] // Ceiling], -1}, 2] // RealDigits[#, 10, digits]& // First (* Jean-François Alcover, Feb 20 2014 *)
1/2-1/4*Product[1-2^(-2^k), {k, 0, Infinity}] // N[#, 105]& // RealDigits // First (* Jean-François Alcover, May 15 2014, after Steven Finch *)
(* ThueMorse function needs $Version >= 10.2 *)
P = FromDigits[{ThueMorse /@ Range[0, 400], 0}, 2];
RealDigits[P, 10, 105][[1]] (* Jean-François Alcover, Jan 30 2020 *)

default(realprecision, 20080); x=0.0; m=67000; for (n=1, m-1, x=x+x; x=x+sum(k=0, length(binary(n))-1, bittest(n, k))%2); x=10*x/2^m; for (n=0, 20000, d=floor(x); x=(x-d)*10; write("b014571.txt", n, " ", d)); \\ Harry J. Smith, Apr 25 2009

1/2-prodinf(n=0,1-1.>>2^n)/4 \\ Charles R Greathouse IV, Jul 31 2012

Equals Sum_{k>=0} A010060(n)*2^(-(k+1)). [Corrected by Jianing Song, Oct 27 2018]
Equals Sum_{k>=1} 2^(-(A000069(k)+1)). - Jianing Song, Oct 27 2018
From Amiram Eldar, Nov 14 2020: (Start)
Equals 1/2 - (1/4) * A215016.
Equals 1/(3 - 1/A247950). (End)

Corrected and extended by R. J. Mathar, Mar 03 2008

A258714 Decimal expansion of Sum_{k>=0} 1/Product_{i=0..k} (2^(2^i) - 1).

Original entry on oeis.org

1, 3, 5, 5, 6, 4, 2, 7, 0, 2, 8, 5, 4, 8, 1, 6, 7, 2, 3, 3, 3, 3, 2, 6, 2, 7, 1, 4, 5, 9, 0, 7, 3, 9, 5, 5, 2, 3, 3, 9, 8, 8, 2, 9, 3, 8, 5, 9, 9, 4, 8, 7, 7, 2, 8, 4, 6, 8, 9, 7, 8, 5, 8, 5, 3, 3, 9, 7, 5, 0, 9, 4, 9, 9, 7, 7, 7, 2, 1, 4

Offset: 1

N. J. A. Sloane, Jun 15 2015

			1.3556427028548167233332627145907395523398829385994877...

Cf. A215016, A258715, A258716, A370458.

RealDigits[1/NProduct[1 - 1/2^(2^k), {k, 0, Infinity}, WorkingPrecision -> 120] - 3/2][[1]] (* Amiram Eldar, Feb 19 2024 *)

1/prodinf(k = 0, 1 - 1/2^(2^k)) - 3/2 \\ Amiram Eldar, Feb 19 2024

From Amiram Eldar, Feb 19 2024: (Start)
Equals Sum_{n>=0} 1/A370458(n).
Equals 1/A215016 - 3/2. (End)

A258716 Decimal expansion of 3 + 2*Sum_{k>=0} 1/Product_{i=0..k} (2^(2^i) - 1).

Original entry on oeis.org

5, 7, 1, 1, 2, 8, 5, 4, 0, 5, 7, 0, 9, 6, 3, 3, 4, 4, 6, 6, 6, 6, 5, 2, 5, 4, 2, 9, 1, 8, 1, 4, 7, 9, 1, 0, 4, 6, 7, 9, 7, 6, 5, 8, 7, 7, 1, 9, 8, 9, 7, 5, 4, 5, 6, 9, 3, 7, 9, 5, 7, 1, 7, 0, 6, 7, 9, 5, 0, 1, 8, 9, 9, 9, 5, 5, 4, 4, 2, 8

Offset: 1

N. J. A. Sloane, Jun 15 2015

			5.7112854057096334466665254291814791046797658771989754...

Gary W. Adamson and N. J. A. Sloane, Correspondence, May 1994, including Adamson's MSS "Algorithm for Generating nth Row of Pascal's Triangle, mod 2, from n", and "The Tower of Hanoi Wheel". Defines this number.

Cf. A215016, A258714, A258715.

RealDigits[2/NProduct[1 - 1/2^(2^k), {k, 0, Infinity}, WorkingPrecision -> 120]][[1]] (* Amiram Eldar, Feb 19 2024 *)

2/prodinf(k = 0, 1 - 1/2^(2^k)) \\ Amiram Eldar, Feb 19 2024

Equals 3 + A258715.
From Amiram Eldar, Feb 19 2024: (Start)
Equals 2 * A258714 + 3.
Equals 2/A215016. (End)

A258715 Decimal expansion of 2*Sum_{k>=0} 1/Product_{i=0..k} (2^(2^i) - 1).

Original entry on oeis.org

2, 7, 1, 1, 2, 8, 5, 4, 0, 5, 7, 0, 9, 6, 3, 3, 4, 4, 6, 6, 6, 6, 5, 2, 5, 4, 2, 9, 1, 8, 1, 4, 7, 9, 1, 0, 4, 6, 7, 9, 7, 6, 5, 8, 7, 7, 1, 9, 8, 9, 7, 5, 4, 5, 6, 9, 3, 7, 9, 5, 7, 1, 7, 0, 6, 7, 9, 5, 0, 1, 8, 9, 9, 9, 5, 5, 4, 4, 2, 8

Offset: 1

N. J. A. Sloane, Jun 15 2015

			2.7112854057096334466665254291814791046797658771989754...

Cf. A215016, A258714, A258716.

RealDigits[2/NProduct[1 - 1/2^(2^k), {k, 0, Infinity}, WorkingPrecision -> 120] - 3][[1]] (* Amiram Eldar, Feb 19 2024 *)

2/prodinf(k = 0, 1 - 1/2^(2^k)) - 3 \\ Amiram Eldar, Feb 19 2024

From Amiram Eldar, Feb 19 2024: (Start)
Equals 2 * A258714.
Equals 2/A215016 - 3. (End)

A370458 Partial products of A051179.

Original entry on oeis.org

1, 3, 45, 11475, 752014125, 3229876072253041875, 59580697294650083747194059426068878125, 20274260698223485458204828871028994444941136941453077244297515184669623921875

Offset: 0

Amiram Eldar, Feb 19 2024

Amiram Eldar, Table of n, a(n) for n = 0..10
Donald Knuth, Problem 11685, The American Mathematical Monthly, Vol. 120, No. 1 (2013), p. 76; The Reciprocal of the Thue-Morse Constant, Solution to Problem 11685 by Traian Viteam, ibid., Vol. 122, No. 1 (2015), pp. 81-82.
Roberto Tauras, Problem 11685.

Cf. A051179, A215016, A258714, A258715, A258716.

FoldList[Times, Table[2^(2^n) - 1, {n, 0, 7}]]

lista(nmax) = {my(v = 1); for(i = 0, nmax, v *= (2^(2^i) - 1); print1(v, ", "));}

from math import prod
def A370458(n): return prod((1<<(1<Chai Wah Wu, Feb 19 2024

a(n) = Product_{k=0..n} A051179(k).

Sum_{n>=0} 1/a(n) = A258714 = 1/A215016 - 3/2 = 1.355642702854... (Knuth, 2013).

A351404 Decimal expansion of Sum_{k>=1} A106400(k-1)/k.

Original entry on oeis.org

3, 9, 8, 7, 6, 1, 0, 8, 8, 1, 0, 8, 4, 1, 8, 8, 1, 2, 4, 0, 7, 4, 3, 0, 5, 4, 4, 4, 0, 0, 2, 7, 3, 0, 6, 0, 3, 3, 6, 8, 0, 8, 9, 1, 5, 4, 6, 7, 1, 9, 8, 1, 2, 7, 2, 9, 9, 5, 7, 4, 4, 4, 5, 7, 6, 9, 2, 7, 9, 1, 7, 2, 0, 3, 6, 3, 8, 6, 0, 2, 9, 2, 5, 9, 7, 0, 6, 5, 4, 8, 4, 8, 7, 3, 7, 0, 2, 4, 9, 6, 5, 4, 4, 4, 3

Offset: 0

Albert Böschow and Julian Böschow, Feb 10 2022

Define S(j) = Sum_{k=1..2^j} A106400(k-1)/k; S(28) agrees with this constant for 104 digits. - Jon E. Schoenfield, Feb 22 2022

			0.39876108810841881240743054440027306033680...

Cf. A106400, A215016.

More digits from Jon E. Schoenfield, Feb 22 2022

A357762 Decimal expansion of -Sum_{k>=1} A106400(k)/k.

Original entry on oeis.org

1, 1, 9, 6, 2, 8, 3, 2, 6, 4, 3, 2, 5, 2, 5, 6, 4, 3, 7, 2, 2, 2, 2, 9, 1, 6, 3, 3, 2, 0, 0, 8, 1, 9, 1, 8, 1, 0, 1, 0, 4, 2, 6, 7, 4, 6, 4, 0, 1, 5, 9, 4, 3, 8, 1, 8, 9, 8, 7, 2, 3, 3, 3, 7, 3, 0, 7, 8, 3, 7, 5, 1, 6, 1, 0, 9, 1, 5, 8, 0, 8, 7, 7, 7, 9, 1, 1, 9, 6, 4, 5, 4, 6, 2, 1, 1, 0, 7, 4, 8, 9, 6, 3, 3, 3

Offset: 1

Amiram Eldar, Oct 12 2022

The asymptotic mean of the excess of the number of odious divisors over the number of evil divisors (A357761, see formula).

The convergence of the partial sums S(m) = -Sum_{k=1..2^m-1} A106400(k)/k is fast: e.g., S(28) is already correct to 100 decimal digits (see also Jon E. Schoenfield's comment in A351404).

			1.19628326432525643722229163320081918101042674640159...

Cf. A106400, A357761.

Similar constants: A215016, A351404

sum = 0; m = 1; pow = 2; Do[sum -= (-1)^DigitCount[k, 2, 1]/k; If[k == pow - 1, Print[m, " ", N[sum, 120]]; m++; pow *= 2], {k, 1, 2^30}]

default(realprecision, 150);
sm = 0.; m = 1; pow = 2; for(k = 1, 2^30, sm -= (-1)^hammingweight(k)/k; if(k == pow - 1, print(m," ",sm); m++; pow *= 2))

Equals -2 * Sum_{k>=1} A106400(2*k-1)/(2*k-1).

Equals lim_{m->oo} (1/m) * Sum_{k=1..m} A357761(k).

A383471 a(n) = round(Product_{k=1..n} (1 + 1/2^2^(-k))).

Original entry on oeis.org

1, 2, 3, 6, 12, 23, 46, 93, 185, 370, 739, 1478, 2955, 5910, 11819, 23637, 47274, 94549, 189097, 378194, 756388, 1512776, 3025551, 6051102, 12102203, 24204407, 48408813, 96817626, 193635251, 387270501, 774541003, 1549082005, 3098164010, 6196328019, 12392656038, 24785312075

Offset: 0

Greg Huber, Apr 27 2025

nonn

Cf. A014571, A215016.

Floor[Product[(1+2^(-2^(-k))),{k,1,n}]+1/2]

a(n) = round(prod(k=1, n, 1 + 1/2^2^(-k))); \\ Michel Marcus, Apr 28 2025

A383472 a(n) = round(Product_{k=1..n} 1 + 2^2^(-k)).

Original entry on oeis.org

1, 2, 5, 11, 23, 46, 92, 184, 369, 738, 1477, 2954, 5909, 11818, 23637, 47274, 94548, 189096, 378193, 756387, 1512775, 3025550, 6051101, 12102203, 24204406, 48408812, 96817625, 193635250, 387270501, 774541002, 1549082004, 3098164009, 6196328018, 12392656037

Offset: 0

Greg Huber, Apr 27 2025

nonn

Cf. A215016, A014571.

Floor[Product[(1+2^(2^(-k))),{k,1,n}]+1/2]

a(n) = round(prod(k=1, n, 1 + 2^2^(-k))); \\ Michel Marcus, Apr 28 2025

cp's OEIS Frontend

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

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A014571 Consider the Morse-Thue sequence (A010060) as defining a binary constant and convert it to decimal.

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A258714 Decimal expansion of Sum_{k>=0} 1/Product_{i=0..k} (2^(2^i) - 1).

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A258716 Decimal expansion of 3 + 2*Sum_{k>=0} 1/Product_{i=0..k} (2^(2^i) - 1).

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A258715 Decimal expansion of 2*Sum_{k>=0} 1/Product_{i=0..k} (2^(2^i) - 1).

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A370458 Partial products of A051179.

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A351404 Decimal expansion of Sum_{k>=1} A106400(k-1)/k.

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