A258714 -id:A258714 - 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)

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).

cp's OEIS Frontend

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

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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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