A132038 -id:A132038 - cp's OEIS frontend

A132325 Decimal expansion of Product_{k>=0} (1+1/10^k).

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

Offset: 1

Hieronymus Fischer, Aug 20 2007

Twice the constant A132326.

			2.22446913827410126425215613418881160749501...

G. C. Greubel, Table of n, a(n) for n = 1..1200
Richard J. McIntosh, Some Asymptotic Formulae for q-Hypergeometric Series, Journal of the London Mathematical Society, Vol. 51, No. 1 (1995), pp. 120-136; alternative link.

Cf. A081845, A067080, A100220, A132019-A132026, A132034-A132038, A132323-A132326, A132271, A132272, A000593.

digits = 105; NProduct[1+1/10^k, {k, 1, Infinity}, NProductFactors -> 100, WorkingPrecision -> digits+3] // N[#, digits+3]& // RealDigits[#, 10, digits]& // First (* Jean-François Alcover, Feb 18 2014 *)
2*N[QPochhammer[-1/10,1/10]] (* G. C. Greubel, Dec 02 2015 *)

prodinf(x=0, 1+(1/10)^x) \\ Altug Alkan, Dec 03 2015

Equals lim sup_{n->oo} Product_{0<=k<=floor(log_10(n))} (1+1/floor(n/10^k)).
Equals lim sup_{n->oo} A132271(n)/n^((1+log_10(n))/2).
Equals lim sup_{n->oo} A132272(n)/n^((log_10(n)-1)/2).
Equals 2*exp(Sum_{n>0} 10^(-n)*Sum_{k|n} -(-1)^k/k) = 2*exp(Sum_{n>0} A000593(n)/(n*10^n)).
Equals lim sup_{n->oo} A132271(n+1)/A132271(n).
Equals 2*(-1/10; 1/10){infinity}, where (a;q){infinity} is the q-Pochhammer symbol. - G. C. Greubel, Dec 02 2015
Equals sqrt(2) * exp(log(10)/24 + Pi^2/(12*log(10))) * Product_{k>=1} (1 - exp(-2*(2*k-1)*Pi^2/log(10))) (McIntosh, 1995). - Amiram Eldar, May 20 2023

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

A270743 Runlength sequence of the zero-one sequence A270742.

Original entry on oeis.org

1, 1, 2, 1, 2, 4, 1, 3, 6, 1, 4, 8, 1, 5, 10, 1, 6, 12, 1, 7, 14, 1, 8, 16, 1, 9, 18, 1, 10, 20, 1, 11, 22, 1, 12, 24, 1, 13, 26, 1, 14, 28, 1, 15, 30, 1, 16, 32, 1, 17, 34, 1, 18, 36, 1, 19, 38, 1, 20, 40, 1, 21, 42, 1, 22, 44, 1, 23, 46, 1, 24, 48, 1, 25

Offset: 1

Clark Kimberling, Apr 04 2016

This is the runlength sequence for C = (1/2)*(3/4)*(7/8)*(15/16)... in each base (2,3,4,...); for base 10, we have C = 0.8900100999989990000001000..., as in A132038.

			(zero-one sequence at A270742) = (0, 1, 0, 0, 1, 0, 0, 1, 1, 1, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0,...), with first 6 runs 0, 1, 00, 1, 00, 1111 of lengths 1,1,2,1,2,4.

Clark Kimberling, Table of n, a(n) for n = 1..1000

Cf. A132038, A270742.

d = Join[{0}, RealDigits[QPochhammer[1/2], 2, 200000][[1]]];
u = Map[Length, Split[d]]

a(3n) = 2n for n > 0, a(3n+1) = 1 for n >= 0, a(3n+2) = n + 1 for n >= 0.

cp's OEIS Frontend

A132325 Decimal expansion of Product_{k>=0} (1+1/10^k).

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

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A270743 Runlength sequence of the zero-one sequence A270742.

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