A247165 -id:A247165 - 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)

A247219 Positive numbers m such that m^2 - 1 divides 2^m - 1.

Original entry on oeis.org

2, 4, 16, 36, 256, 456, 1296, 2556, 4356, 6480, 8008, 11952, 26320, 44100, 47520, 47880, 49680, 57240, 65536, 74448, 84420, 97812, 141156, 157080, 165600, 225456, 278496, 310590, 333432, 365940, 403900, 419710, 476736, 557040, 560736, 576720, 647088, 1011960, 1033056, 1204560, 1206180

Offset: 1

Juri-Stepan Gerasimov, Nov 26 2014

nonn

Contains all numbers of the form m = A001146(k) = 2^2^k, k >= 0; and those with k > 1 seem to form the intersection with A247165. - M. F. Hasler, Jul 25 2015

			2 is in this sequence because 2^2 - 1 = 3 divides 2^2 - 1 = 3.

Chai Wah Wu, Table of n, a(n) for n = 1..105

Cf. A081762.

[n: n in [2..122222] | Denominator((2^n - 1)/(n^2 - 1)) eq 1];

Select[Range[10^4], Divisible[2^# - 1, #^2 - 1] &] (* Alonso del Arte, Nov 26 2014 *)
Select[Range[2,121*10^4],PowerMod[2,#,#^2-1]==1&] (* Harvey P. Dale, Sep 08 2021 *)

isok(n) = ((2^n - 1) % (n^2 - 1)) == 0; \\ Michel Marcus, Nov 26 2014

forstep(n=0,1e8,2, Mod(2,n^2-1)^n-1 || print1(n", ")) \\ M. F. Hasler, Jul 25 2015

from gmpy2 import powmod
A247219_list = [n for n in range(2,10**7) if powmod(2,n,n*n-1) == 1]
# Chai Wah Wu, Dec 03 2014

Corrected a(24) by Chai Wah Wu, Dec 03 2014

A247220 Numbers k such that k^2 + 1 divides 2^k + 1.

Original entry on oeis.org

0, 2, 4, 386, 20136, 59140, 373164544

Offset: 1

Juri-Stepan Gerasimov, Nov 26 2014

a(8) > 10^12. - Giovanni Resta, May 05 2020
All terms of the sequence are even. a(5), a(6) and a(7) are of the form 2*p + 2 where p is a prime and p mod 14 = 1. - Farideh Firoozbakht, Dec 07 2014
From Jianing Song, Jan 13 2019: (Start)
Among the known terms only a(3) and a(4) are of the form 2*p where p is a prime.
a(n)^2 + 1 is prime for 2 <= n <= 7. Among these primes, the multiplicative order of 2 modulo a(n)^2 + 1 is 2*a(n) except for n = 5, in which case it is 2*a(n)/3. (End)
If a(n)^2 + 1 is composite, then a(n) is also a term of A135590. - Max Alekseyev, Apr 25 2024

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

Cf. A135590, A247165.

for(n=0,10^5,if(Mod(2,n^2+1)^n==-1,print1(n,", "))); \\ Joerg Arndt, Nov 30 2014

from gmpy2 import powmod
A247220_list = [i for i in range(10**7) if powmod(2,i,i*i+1) == i*i]
# Chai Wah Wu, Dec 03 2014

a(7) from Hiroaki Yamanouchi, Nov 29 2014

A260407 Numbers n such that (n-1)^2+1 divides 2^(n-1)-1.

Original entry on oeis.org

1, 17, 257, 8209, 65537, 649801, 1382401, 4294967297

Offset: 1

M. F. Hasler, Jul 24 2015

a(7) = 1382401 is the first composite number of this sequence (which makes it different from A260072).
The Fermat numbers 2^(2^k)+1 = A000215(k) with k>1 are a subsequence of this sequence. I conjecture that they are equal to the intersection of this and A260406 (except for the conventional 1).
Conjecture: also numbers n such that ((2^k)^(n-1)-1) == 0 mod ((n-1)^2+1) for all k >= 1. - Jaroslav Krizek, Jun 02 2016

Cf. A000215, A081762, A247165, A260072, A260406.

[n: n in [1..10^6] | (2^(n-1)-1) mod ((n-1)^2+1) eq 0 ]; // Vincenzo Librandi, Jul 25 2015

Join [{1},Select[Range[43*10^8],PowerMod[2,#-1,(#-1)^2+1]==1&]] (* Harvey P. Dale, Sep 07 2018 *)

forstep(n=1,1e7,2,Mod(2,(n-1)^2+1)^(n-1)==1&&print1(n","))

a(n) = A247165(n)+1.

cp's OEIS Frontend

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

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A247219 Positive numbers m such that m^2 - 1 divides 2^m - 1.

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A247220 Numbers k such that k^2 + 1 divides 2^k + 1.

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A260407 Numbers n such that (n-1)^2+1 divides 2^(n-1)-1.

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Author

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Crossrefs

Programs

Magma

Mathematica

PARI

Formula