A001146 -id:A001146 - cp's OEIS frontend

A215016 Decimal expansion of the product of 1 - 1/2^2^n over all n >= 0.

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

Offset: 0

Charles R Greathouse IV, Jul 31 2012

Can be used to efficiently compute A014571: A014571 = 1/2 - (1/4) * A215016.

			0.35018386543956960886655452696617886764208650217692176970648233860482563...

Joerg Arndt, Matters Computational (The Fxtbook), p.727 (rel. 38.1-5).
R. Schroeppel and R. W. Gosper, HACKMEM #122 (1972).

Cf. A001146, A014571, A106400, A258714, A258716.

RealDigits[NProduct[1 - 1/2^2^n, {n, 0, Infinity}, WorkingPrecision -> 120]][[1]] (* Alonso del Arte, Jul 31 2012 *)

PARI
```
prodinf(n=0,1-1.>>2^n)
```

Equals Sum_{n>=0} A106400(n)/2^n. - Robert FERREOL, Jan 10 2022
From Amiram Eldar, Feb 19 2024: (Start)
Equals Product_{n>=0} (1 - 1/A001146(n)).
Equals 2/A258716.
Equals 1/(3/2 + A258714). (End)

A334866 a(0) = 1, and then after, a(2n) = a(n)^2, a(2n+1) = A334747(a(n)).

Original entry on oeis.org

1, 2, 4, 3, 16, 8, 9, 6, 256, 32, 64, 12, 81, 18, 36, 5, 65536, 512, 1024, 48, 4096, 128, 144, 24, 6561, 162, 324, 27, 1296, 72, 25, 10, 4294967296, 131072, 262144, 768, 1048576, 2048, 2304, 96, 16777216, 8192, 16384, 192, 20736, 288, 576, 20, 43046721, 13122, 26244, 243, 104976, 648, 729, 54, 1679616, 2592, 5184, 108, 625, 50, 100, 15

Offset: 0

Antti Karttunen, Jun 08 2020

This irregular table can be represented as a binary tree. Each child to the left is obtained by squaring the parent, and each child to the right is obtained by applying A334747 to the parent:
                                        1
                                        |
                     ...................2...................
                    4                                       3
         16......../ \........8                   9......../ \........6
         / \                 / \                 / \                 / \
        /   \               /   \               /   \               /   \
       /     \             /     \             /     \             /     \
    256       32         64       12         81       18         36       5
65536  512 1024 48   4096  128 144  24   6561  162 324  27   1296  72   25 10
etc.
This is the mirror image of the tree in A334860.

Index entries for sequences that are permutations of the natural numbers

Cf. A334865 (inverse permutation), A334860 (mirror image).
Composition of permutations A005940 and A225546.
Cf. A000290, A048675, A087808, A334747, A334870.
Cf. A001146 (left edge of the tree), A019565 (right edge), A334110 (the left children of the right edge).

A334747(n) = { my(c=core(n), m=n); forprime(p=2, , if(c % p, m*=p; break, m/=p)); m; }; \\ From A334747
A334866(n) = if(!n,1,if(!(n%2),A334866(n/2)^2,A334747(A334866((n-1)/2))));

a(0) = 1, and then after, a(2n) = a(n)^2, a(2n+1) = A334747(a(n)).
a(n) = A225546(A005940(1+n)).
For all n >= 0, A048675(a(n)) = A087808(n).

A055777 a(n) = 3^(3^n).

Original entry on oeis.org

3, 27, 19683, 7625597484987, 443426488243037769948249630619149892803

Offset: 0

Henry Bottomley, Jul 12 2000

nonn

Next term is too big to include.
a(n+1) = a(n) written in base 3 and read as if in base 27 (and recorded in base 10).
Number of distinct n-ary operators in a ternary logic. - Ross Drewe, Feb 13 2008
The next term has 116 digits. - Harvey P. Dale, Mar 28 2019

Amiram Eldar, Table of n, a(n) for n = 0..6

Cf. A001146, A023365, A383817.

NestList[#^3&,3,5] (* Harvey P. Dale, Mar 28 2019 *)

print([3**(3**n) for n in range(5)]) # Michael S. Branicky, Mar 27 2021

a(n) = a(n-1)^3.

Sum_{n>=0} 1/a(n) = A383817. - Amiram Eldar, May 16 2025

A100441 a(n) is the denominator of f(n) where f(1) = 2 and f(n+1) is the solution of x + Sum_{i=1..n} f(i) = x * Product_{i=1..n} f(i).

Original entry on oeis.org

1, 1, 3, 13, 217, 57073, 3811958497, 16605534578235736513, 309708098978072051970763989442580255617, 106322990835084829467725909226560893968664147958670035553130958199430801942273

Offset: 1

Gilbert Boily (sgbl(AT)escape.ca), Nov 21 2004, Sep 03 2007

Let E(0) = x + 1, let E(n+1) = 1 - E(n) + E(n)^2. Let e(n) = discrim(E(n),x) and let f(n) = e(n+1)/e(n)^2. Then f(1,2,3,...) = -3,13,217,57073,381195849,... which looks like this sequence (I do not have a proof yet). - Daniel R. L. Brown (dbrown(AT)certicom.com), Nov 18 2005
This sequence gives the next number in a sequence where the sum and the product of the terms of the sequence are equal.
It happens that the sum or product of the terms of this sequence match A001146 for the numerator of the sum or product and A076628 for the denominator of the sum or product of the sequence.
Let g(x) = x^2 - x + 1 be the map producing Sylvester's sequence A000058. Then for n >= 0, g^n(1/2) = 1/f(n+2), where g^n is the n-th iterate of g, so a(n+2) is the numerator of g^n(1/2). - Curtis Bechtel, Apr 05 2024

			2, 2, 4/3, 16/13, 256/217, 65536/57073, 4294967296/3811958497, 18446744073709551616/16605534578235736513, ... = A001146/A100441 (essentially).

N. MacKinnon and N. Lord, Sums equal to products, The Mathematical Gazette, March 1986, 21-22.
Crux Mathematicorum, Mathematical mayhem pb no. 114, Vol 30, 2004, p. 467-468. [_Robert FERREOL_, Jul 06 2015]

Cf. A001146, A076628.

I:=[1,3]; [1] cat  [n le 2 select I[n] else 2^(2^(n-1))-2^(2^(n-2))*Self(n-1)+Self(n-1)^2: n in [1..10]]; // Vincenzo Librandi, Jun 13 2015

f:=proc(n) option remember; local i,k,k1,k2; if n = 1 then return(2); fi; k:=mul(f(i),i=1..n-1); k1:=numer(k); k2:=denom(k); k1/(k1-k2); end;
f:=n-> if n=1 or n=2 then 2 else f(n-1)^2/(f(n-1)^2-f(n-1)+1) fi; # Robert FERREOL, Jun 12 2015

f[n_] := f[n] = (frac = Product[f[i], {i, 1, n-1}]; p = Numerator[frac]; q = Denominator[frac]; p/(p-q)); f[1] = 2; (* or, after Robert FERREOL *) f[n_] := f[n] = If[n == 1 || n == 2, 2, f[n-1]^2/(f[n-1]^2-f[n-1]+1)]; Table[f[n], {n, 1, 10}] // Denominator (* Jean-François Alcover, Sep 19 2012, updated Jun 15 2015 *)

{a(n) = my(s, t); if( n<3, n>0, t = a(n-1); s = 2^(2^(n-3)); s*s -s*t +t*t)}; /* Michael Somos, Aug 05 2017 */

@CachedFunction
def a(n): # a = A100441
    if (n<3): return 2*n-1
    else: return 2^(2^(n-1)) - 2^(2^(n-2))*a(n-1) + a(n-1)^2
[1]+[a(n) for n in range(1,12)] # G. C. Greubel, Apr 08 2023

Let F(n) = Product_{i=1..n} f(i) = p/q (say). Then f(n+1) = p/(p-q).
From Robert FERREOL, Jun 12 2015: (Start)
Recurrence: f(1) = f(2) = 2; f(n+1) = f(n)^2/(f(n)^2 - f(n) + 1).
Since f(n) = 2^(2^(n-2))/a(n) for n >= 2, the recurrence for a(n) is:
a(1) = a(2) = 1; a(n+1) = 2^(2^(n-1)) - 2^(2^(n-2))*a(n) + a(n)^2.
(End)

Name edited by Michael Somos, Aug 05 2017

A318128 Number of set-systems (finite sets of finite nonempty sets) with union {1,2,...,n} and intersection {}.

Original entry on oeis.org

1, 0, 2, 84, 31478, 2147000136, 9223371998203475474, 170141183460469231537996491257596836636, 57896044618658097711785492504343953922551603929769020459976077632195066756398

Offset: 0

Gus Wiseman, Aug 18 2018

nonn

			The a(2) = 2 set-systems are {{1},{2}}, and {{1},{2},{1,2}}.

Cf. A000371, A001146, A003465, A119563, A131288.

Cf. A318128, A318129, A318130, A318131, A318132.

Table[Length[Select[Subsets[Rest[Subsets[Range[n]]]],And[Union@@#===Range[n],Intersection@@#=={}]&]],{n,4}]

Inverse binomial transform of A318129.

A334860 a(0) = 1, a(1) = 2, after which, a(2n) = A334747(a(n)), a(2n+1) = a(n)^2.

Original entry on oeis.org

1, 2, 3, 4, 6, 9, 8, 16, 5, 36, 18, 81, 12, 64, 32, 256, 10, 25, 72, 1296, 27, 324, 162, 6561, 24, 144, 128, 4096, 48, 1024, 512, 65536, 15, 100, 50, 625, 108, 5184, 2592, 1679616, 54, 729, 648, 104976, 243, 26244, 13122, 43046721, 20, 576, 288, 20736, 192, 16384, 8192, 16777216, 96, 2304, 2048, 1048576, 768, 262144, 131072, 4294967296, 30

Offset: 0

Antti Karttunen, Jun 08 2020

This irregular table can be represented as a binary tree. Each child to the left is obtained by applying A334747 to the parent, and each child to the right is obtained by squaring the parent:
                                      1
                                      |
                   ...................2...................
                  3                                       4
        6......../ \........9                   8......../ \........16
       / \                 / \                 / \                 / \
      /   \               /   \               /   \               /   \
     /     \             /     \             /     \             /     \
    5       36         18       81         12       64         32      256
  10 25   72  1296   27  324 162  6561   24  144  128 4096   48 1024 512 65536
etc.
This is the mirror image of the tree in A334866.
Fermi-Dirac primes, A050376, occur at rightward growing branches that originate from primes situated at the left edge.
The tree illustrated in A163511 is expanded as x -> 2*x for the left child and x -> A003961(x) for the right child, while this tree is expanded as x -> A225546(2*A225546(x)) for the left child, and x -> A225546(A003961(A225546(x))) for the right child.

Index entries for sequences that are permutations of the natural numbers

Cf. A000290, A225546, A334204, A334747, A334859 (inverse), A334866 (mirror image).
Cf. A001146 (right edge of the tree), A019565 (left edge), A334110 (the right children of the left edge).
Composition of permutations A163511 and A225546.

A334747(n) = { my(c=core(n), m=n); forprime(p=2, , if(c % p, m*=p; break, m/=p)); m; }; \\ From A334747
A334860(n) = if(n<=1,1+n,if(!(n%2),A334747(A334860(n/2)),A334860((n-1)/2)^2));

a(0) = 1, a(1) = 2; and for n > 0, a(2n) = A334747(a(n)), a(2n+1) = a(n)^2.
a(n) = A225546(A163511(n)).
For n >= 0, a(2^n) = A019565(1+n), a(2^((2^n)-1)) = A000040(1+n).
A334109(a(n)) = A334204(n).
It seems that for n >= 1, A048675(a(n)) = A135529(n) = A048675(A163511(n)).

A057156 Number of functions from {0,1}^n to {0,1}^n.

Original entry on oeis.org

1, 4, 256, 16777216, 18446744073709551616, 1461501637330902918203684832716283019655932542976, 39402006196394479212279040100143613805079739270465446667948293404245721771497210611414266254884915640806627990306816

Offset: 0

Henry Bottomley, Aug 15 2000

a(n) is the number of subdivisions of the Brownian motion on the unit interval at the n-th stage of subdivision. - Stephen Crowley, Apr 12 2007

			a(1)=4 since the possibilities are f(0)=0, f(1)=0; f(0)=0, f(1)=1; f(0)=1, f(1)=0; f(0)=1, f(1)=1.
For n=3: we need to count maps from a set with 8 points to a set with 8 points.  There are 8^8 such functions, that is, a(3) = 8^8 = 2^24 = 16777216. - _N. J. A. Sloane_, Mar 05 2023

François Robert, Discrete Iterations: A Metric Study, Springer-Verlag, 1986, p. 167.
Norbert Wiener, Nonlinear Problems in Random Theory, MIT Press Classic, 1958, Lecture 1.

Alois P. Heinz, Table of n, a(n) for n = 0..8

Cf. A000079, A000312, A000722, A001146, A036289, A043322, A057157, A092258, A134880.

lst={};Do[AppendTo[lst,(2^n)^(2^n)],{n,0,8}];lst (* Vladimir Joseph Stephan Orlovsky, Mar 02 2009 *)

a(n)=1<<(n<Charles R Greathouse IV, Jan 19 2012

a(n) = (2^n)^(2^n) = A000312(A000079(n)) = A000079(A036289(n)) = A001146(n)^n = A000722(n) + A057157(n).

Sum_{n>=1} 1/a(n) = A134880. - Amiram Eldar, Nov 15 2020

More terms from Vladimir Joseph Stephan Orlovsky, Mar 02 2009

A176594 a(n) = 5^(2^n).

Original entry on oeis.org

5, 25, 625, 390625, 152587890625, 23283064365386962890625, 542101086242752217003726400434970855712890625, 293873587705571876992184134305561419454666389193021880377187926569604314863681793212890625

Offset: 0

Vincenzo Librandi, Apr 21 2010

nonn

Also the hypotenuse of primitive Pythagorean triangles obtained by repeated application of basic formula c(n)=p(n)^2+q(n)^2 starting p(0)=2, q(0)=1, see A100686, A098122. Example: a(2)=25 since starting (2,1) gives Pythagorean triple (3,4,5) using (3,4) as new generators gives triple (7,24,25) hypotenuse 25=a(2). - Carmine Suriano, Feb 04 2011

Paolo Xausa, Table of n, a(n) for n = 0..10

Cf. A001146, A011764, A078886, A098122, A100686, A165423.

Cf. A185457, A120905, A139011. - Carmine Suriano, Feb 04 2011

NestList[#^2 &, 5, 8] (* Paolo Xausa, Jul 13 2025 *)

a(n) = 5^(2^n); \\ Michel Marcus, Jan 26 2016

a(n) = A165423(n+3).
a(n+1) = a(n)^2 with a(0)=5.
a(n-1) = (Im((2+i)^(2^n))^2 + Re((2+i)^(2^n))^2)^(1/2). - Carmine Suriano, Feb 04 2011
Sum_{n>=0} 1/a(n) = A078886. - Amiram Eldar, Nov 09 2020
Product_{n>=0} (1 + 1/a(n)) = 5/4. - Amiram Eldar, Jan 29 2021

Offset corrected by R. J. Mathar, Jun 18 2010

A318129 Number of sets of nonempty subsets of {1,...,n} with intersection {}.

Original entry on oeis.org

1, 1, 3, 91, 31827, 2147158387, 9223372011085950171, 170141183460469231602560095290109272523, 57896044618658097711785492504343953923912733397452774312538303978325772978595

Offset: 0

Gus Wiseman, Aug 18 2018

nonn

			The a(2) = 3 sets of sets are {}, {{1},{2}}, {{1},{2},{1,2}}.

Cf. A000371, A001146, A003465,A040996, A119563, A131288.

Cf. A318128, A318130, A318131, A318132.

Table[Length[Select[Subsets[Rest[Subsets[Range[n]]]],Or[#=={},Intersection@@#=={}]&]],{n,0,4}]

Binomial transform of A318128.

a(n) = A318130(n) - 2^(2^n - 1). [corrected]

A369938 Numbers whose maximal exponent in their prime factorization is a power of 2.

Original entry on oeis.org

2, 3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 25, 26, 28, 29, 30, 31, 33, 34, 35, 36, 37, 38, 39, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 55, 57, 58, 59, 60, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 73, 74, 75, 76, 77

Offset: 1

Amiram Eldar, Feb 06 2024

First differs from its subsequence A138302 \ {1} at n = 378: a(378) = 432 = 2^4 * 3^3 is not a term of A138302.
First differs from A096432, A220218 \ {1}, A335275 \ {1} and A337052 \ {1} at n = 56, and from A270428 \ {1} at n = 113.
Numbers k such that A051903(k) is a power of 2.
The asymptotic density of this sequence is 1/zeta(3) + Sum_{k>=2} (1/zeta(2^k+1) - 1/zeta(2^k)) = 0.87442038669659566330... .

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000079, A002117, A051903.
Subsequences: A001146, A001248, A005117, A030514, A030635, A050376, A062503, A113849, A138302 \ {1}, A179645.
Similar sequences: A368714, A369937, A369939.
Cf. A096432, A220218, A270428, A335275, A337052.

pow2Q[n_] := n == 2^IntegerExponent[n, 2];
Select[Range[2, 100], pow2Q[Max[FactorInteger[#][[;; , 2]]]] &]
Select[Range[2,80],IntegerQ[Log2[Max[FactorInteger[#][[;;,2]]]]]&] (* Harvey P. Dale, Nov 06 2024 *)

ispow2(n) = n >> valuation(n, 2) == 1;
is(n) = n > 1 && ispow2(vecmax(factor(n)[, 2]));

cp's OEIS Frontend

A215016 Decimal expansion of the product of 1 - 1/2^2^n over all n >= 0.

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A334866 a(0) = 1, and then after, a(2n) = a(n)^2, a(2n+1) = A334747(a(n)).

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A055777 a(n) = 3^(3^n).

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A100441 a(n) is the denominator of f(n) where f(1) = 2 and f(n+1) is the solution of x + Sum_{i=1..n} f(i) = x * Product_{i=1..n} f(i).

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A318128 Number of set-systems (finite sets of finite nonempty sets) with union {1,2,...,n} and intersection {}.

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A334860 a(0) = 1, a(1) = 2, after which, a(2n) = A334747(a(n)), a(2n+1) = a(n)^2.

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A057156 Number of functions from {0,1}^n to {0,1}^n.

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