A132323 -id:A132323 - cp's OEIS frontend

A323716 a(n) = Product_{k=0..n} (3^k + 1).

2, 8, 80, 2240, 183680, 44817920, 32717081600, 71584974540800, 469740602936729600, 9246374028206585446400, 545998386365598870609920000, 96722522147893108730806108160000, 51402410615320609490117059732766720000

Offset: 0

Vaclav Kotesovec, Jan 25 2019

nonn

G. C. Greubel, Table of n, a(n) for n = 0..64

Cf. A028361, A034472, A132323, A323715.

[(&*[3^j+1: j in [0..n]]): n in [0..20]]; // G. C. Greubel, Aug 30 2023

Table[Product[3^k+1, {k, 0, n}], {n, 0, 12}]
Table[QPochhammer[-1, 3, n+1], {n, 0, 12}]

a(n) = prod(k=0, n, 3^k+1); \\ Michel Marcus, Jan 25 2019

[product(3^j+1 for j in range(n+1)) for n in range(21)] # G. C. Greubel, Aug 30 2023

a(n) ~ c * 3^(n*(n+1)/2), where c = A132323 = QPochhammer[-1, 1/3] = 3.12986803713402307587769821345767...

A371748 Decimal expansion of Product_{k>=0} (1 + 1/4^k).

Original entry on oeis.org

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

Offset: 1

Ilya Gutkovskiy, Apr 05 2024

			2.71181934772695876069108846970791860244...

Kelvin Voskuijl, Table of n, a(n) for n = 1..20000

Cf. A000302, A065445, A081845, A100221, A132323, A371747, A371749, A371751.

RealDigits[QPochhammer[-1, 1/4], 10, 100][[1]]

Equals A065445^2. - Hugo Pfoertner, Apr 05 2024

A371751 Decimal expansion of Product_{k>=0} (1 + 1/5^k).

Original entry on oeis.org

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

Offset: 1

Ilya Gutkovskiy, Apr 05 2024

			2.5210016134202150647777463154782130132...

Kelvin Voskuijl, Table of n, a(n) for n = 1..20000

Cf. A000351, A081845, A100222, A132323, A371748, A371750, A371752.

RealDigits[QPochhammer[-1, 1/5], 10, 100][[1]]
RealDigits[Times@@Table[1+1/5^k,{k,0,1000}],10,100][[1]] (* Harvey P. Dale, Sep 17 2024 *)

A132266 Decimal expansion of Product_{k>=0} (1 - 1/(2*12^k)).

Original entry on oeis.org

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

Offset: 0

Hieronymus Fischer, Aug 20 2007

			0.47735217025489380198334286365820...

G. C. Greubel, Table of n, a(n) for n = 0..2000

Cf. A000217, A048651, A132019-A132026, A132034-A132038, A132265-A132268, A132323-A132326, A132264.

digits = 105; NProduct[1-1/(2*12^k), {k, 0, Infinity}, NProductFactors -> 100, WorkingPrecision -> digits+5] // N[#, digits+5]& // RealDigits[#, 10, digits]& // First (* Jean-François Alcover, Feb 18 2014 *)
(1/2)*N[QPochhammer[1/24, 1/12], 200] (* G. C. Greubel, Dec 20 2015 *)

prodinf(x=0, 1-1/(2*12^x)) \\ Altug Alkan, Dec 20 2015

lim inf (Product_{k=0..floor(log_12(n))} floor(n/12^k)*12^k/n) for n-->oo.
lim inf A132264(n)*12^((1+floor(log_12(n)))*floor(log_12(n))/2)/n^(1+floor(log_12(n))) for n-->oo.
lim inf A132264(n)*12^A000217(floor(log_12(n)))/n^(1+floor(log_12(n))) for n-->oo.
(1/2)*exp(-Sum_{n>0} 12^(-n)*Sum_{k|n} 1/(k*2^k)).
lim inf A132264(n)/A132264(n+1) = 0.47735217025489380... for n-->oo.
Equals (1/2)*(1/24; 1/12){infinity}, where (a;q){infinity} is the q-Pochhammer symbol. - G. C. Greubel, Dec 20 2015

A371746 Decimal expansion of Product_{k>=0} 1 / (1 + 1/3^k).

Original entry on oeis.org

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

Offset: 0

Ilya Gutkovskiy, Apr 05 2024

			0.31950228831873889019480071011090065424...

Cf. A000244, A083864, A100220, A132323, A370466, A371749, A371752.

RealDigits[1/QPochhammer[-1, 1/3], 10, 100][[1]]

Equals 1 / A132323.

cp's OEIS Frontend

A323716 a(n) = Product_{k=0..n} (3^k + 1).

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A371748 Decimal expansion of Product_{k>=0} (1 + 1/4^k).

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A371751 Decimal expansion of Product_{k>=0} (1 + 1/5^k).

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A132266 Decimal expansion of Product_{k>=0} (1 - 1/(2*12^k)).

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A371746 Decimal expansion of Product_{k>=0} 1 / (1 + 1/3^k).

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