A247219 -id:A247219 - 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)

A247165 Numbers m such that m^2 + 1 divides 2^m - 1.

Original entry on oeis.org

0, 16, 256, 8208, 65536, 649800, 1382400, 4294967296

Offset: 1

Juri-Stepan Gerasimov, Nov 30 2014

Contains 2^(2^k) = A001146(k) for k >= 2. - Robert Israel, Dec 02 2014
a(9) > 10^12. - Hiroaki Yamanouchi, Sep 16 2018
For each n, a(n)^2 + 1 belongs to A176997, and thus a(n) belongs to either A005574 or A135590. - Max Alekseyev, Feb 08 2024

			0 is in this sequence because 0^2 + 1 = 1 divides 2^0 - 1 = 1.

Cf. A247219 (n^2 - 1 divides 2^n - 1), A247220 (n^2 + 1 divides 2^n + 1).

Cf. A005574, A135590, A176997.

[n: n in [1..100000] | Denominator((2^n-1)/(n^2+1)) eq 1];

select(n -> (2 &^ n - 1) mod (n^2 + 1) = 0, [$1..10^6]); # Robert Israel, Dec 02 2014

a247165[n_Integer] := Select[Range[0, n], Divisible[2^# - 1, #^2 + 1] &]; a247165[1500000] (* Michael De Vlieger, Nov 30 2014 *)

for(n=0,10^9,if(Mod(2,n^2+1)^n==+1,print1(n,", "))); \\ Joerg Arndt, Nov 30 2014

A247165_list = [n for n in range(10**6) if n == 0 or pow(2,n,n*n+1) == 1]
# Chai Wah Wu, Dec 04 2014

a(8) from Chai Wah Wu, Dec 04 2014

Edited by Jon E. Schoenfield, Dec 06 2014

A271842 Positive numbers m such that m^2 - 1 divides 4^m - 1.

Original entry on oeis.org

2, 4, 6, 16, 36, 52, 66, 256, 378, 456, 1296, 1470, 1548, 1800, 2002, 2556, 4356, 6480, 8008, 11952, 23580, 26320, 33930, 36636, 37170, 43290, 44100, 47520, 47880, 49680, 57240, 65536, 74448, 84420, 97812, 101920, 127050, 134946, 139860, 141156, 157080, 164880, 165600, 209220, 225456

Offset: 1

Juri-Stepan Gerasimov, Apr 15 2016

nonn

			2 is in this sequence because (4^2 - 1)/(2^2 - 1) = 5.

Charles R Greathouse IV, Table of n, a(n) for n = 1..1000

Cf. positive numbers m such that m^2 - 1 divides (2^k)^m - 1:

A247219 (k=1), this sequence (k=2), A242062 (k=3).

[0] cat [n: n in [2..240000] | Denominator((4^n-1)/(n^2-1)) eq 1];

A271842:=n->`if`((4^n-1) mod (n^2-1) = 0, n, NULL): seq(A271842(n), n=2..10^4); # Wesley Ivan Hurt, Apr 18 2016

Select[Range[1, 100], IntegerQ[(4^# - 1)/(#^2 - 1)] &] (* G. C. Greubel, Apr 15 2016 *)

is(n)=Mod(4,n^2-1)^n==1 \\ Charles R Greathouse IV, Apr 15 2016

A272062 Positive numbers k such that k^2 - 1 divides 8^k - 1.

Original entry on oeis.org

2, 4, 8, 10, 16, 22, 36, 40, 64, 96, 100, 196, 210, 256, 280, 316, 456, 560, 820, 1200, 1236, 1296, 1360, 1408, 1600, 1870, 2380, 2556, 3516, 3616, 4096, 4200, 4356, 5656, 6112, 6256, 6480, 8008, 8688, 10192, 10356, 11440, 11952, 12160, 13728, 14950, 16192, 17020, 19432, 21880, 22036

Offset: 1

Juri-Stepan Gerasimov, Apr 19 2016

nonn

From Robert Israel, Jun 08 2018: (Start)
All terms are even.
Are 2, 8 and 560 the only terms == 2 (mod 6)?  There are no others up to 3*10^9. (End)

			a(1) = 2 because (8^2 - 1)/(2^2 - 1) = 21.

Robert Israel, Table of n, a(n) for n = 1..2559

Cf. positive numbers n such that n^2 - 1 divides (2^k)^n - 1: A247219 (k=1), A271842 (k=2), this sequence (k=3).

[0] cat [n: n in [2..30000] | Denominator((8^n-1)/(n^2-1)) eq 1];

A272062:=n->`if`((8^n-1) mod (n^2-1) = 0, n, NULL): seq(A272062(n), n=2..5*10^4); # Wesley Ivan Hurt, Apr 21 2016

Select[Range[2, 22100], Divisible[8^# - 1, #^2 - 1] &] (* Michael De Vlieger, Apr 19 2016 *)

is(n)=Mod(8,n^2-1)^n==1 \\ Charles R Greathouse IV, Apr 19 2016

A281363 Smallest m>0 such that (2n)^2 - 1 divides (2^m)^(2n) - 1.

Original entry on oeis.org

1, 1, 2, 3, 3, 5, 6, 1, 4, 9, 3, 55, 90, 9, 14, 5, 30, 1, 18, 3, 10, 21, 6, 161, 84, 2, 130, 45, 9, 29, 30, 3, 2, 33, 11, 35, 90, 15, 5, 351, 27, 82, 28, 7, 22, 15, 90, 3, 120, 3, 50, 51, 6, 53, 18, 9, 154, 33, 12, 11, 110, 25, 50, 7, 7, 195, 18, 9, 34, 69

Offset: 1

Juri-Stepan Gerasimov, Apr 30 2016

nonn

			a(3) = 2 because (2*3)^2 - 1 = 35 divides (2^2)^(2*3) - 1 = 4095.

Giovanni Resta and Chai Wah Wu, Table of n, a(n) for n = 1..10000 (terms for n = 1..1000 from Giovanni Resta)

Cf. positive numbers n such that n^2 - 1 divides (2^k)^n - 1: A247219 (k=1), A271842 (k=2), A272062 (k=3).

Table[SelectFirst[Range@ 1200, Divisible[(2^#)^(2 n) - 1, (2 n)^2 - 1] &], {n, 84}] (* Michael De Vlieger, May 01 2016, Version 10 *)
a[n_] := Block[{m=1}, While[ PowerMod[2^m, 2*n, 4*n^2-1] != 1, m++]; m]; Array[a, 100] (* Giovanni Resta, May 05 2016 *)

def A281363(n):
    m, q = 1, 4*n**2-1
    p = pow(2, 2*n, q)
    r = p
    while r != 1:
        m += 1
        r = (r*p) % q
    return m # Chai Wah Wu, Jan 28 2017

cp's OEIS Frontend

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

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A247165 Numbers m such that m^2 + 1 divides 2^m - 1.

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A271842 Positive numbers m such that m^2 - 1 divides 4^m - 1.

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A272062 Positive numbers k such that k^2 - 1 divides 8^k - 1.

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A281363 Smallest m>0 such that (2*n)^2 - 1 divides (2^m)^(2*n) - 1.

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A247205 Numbers k such that 2*k^2 - 1 divides 2^k - 1.

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A281363 Smallest m>0 such that (2n)^2 - 1 divides (2^m)^(2n) - 1.