A098844 -id:A098844 - cp's OEIS frontend

A132023 Decimal expansion of Product_{k>=0} 1-1/(2*7^k).

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

Offset: 0

Hieronymus Fischer, Aug 14 2007

			0.4587667266997689850200...

Cf. A048651, A098844, A067080, A132019, A132026, A132031, A132035, A000217, A109808.

digits = 103; NProduct[1-1/(2*7^k), {k, 0, Infinity}, NProductFactors -> 200, WorkingPrecision -> digits+5] // N[#, digits+5]& // RealDigits[#, 10, digits]& // First (* Jean-François Alcover, Feb 18 2014 *)
RealDigits[QPochhammer[1/2, 1/7], 10, 120][[1]] (* Amiram Eldar, May 08 2023 *)

Equals lim inf_{n->oo} Product_{k=0..floor(log_7(n))} floor(n/7^k)*7^k/n.
Equals lim inf_{n->oo} A132031(n)/n^(1+floor(log_7(n)))*7^(1/2*(1+floor(log_7(n)))*floor(log_7(n))).
Equals lim inf_{n->oo} A132031(n)/n^(1+floor(log_7(n)))*7^A000217(floor(log_7(n))).
Equals 1/2*exp(-Sum_{n>0} 7^(-n)*Sum_{k|n} 1/(k*2^k)).
Equals lim inf_{n->oo} A132031(n)/A132031(n+1).
Equals Product_{n>=1} (1 - 1/A109808(n)). - Amiram Eldar, May 08 2023

A132025 Decimal expansion of Product_{k>=0} 1-1/(2*9^k).

Original entry on oeis.org

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

Offset: 0

Hieronymus Fischer, Aug 14 2007

			0.4689451783670236932832800...

Cf. A048651, A098844, A067080, A132019, A132026, A132033, A132037, A000217, A270369.

digits = 103; NProduct[1-1/(2*9^k), {k, 0, Infinity}, NProductFactors -> 100, WorkingPrecision -> digits+5] // N[#, digits+5]& // RealDigits[#, 10, digits]& // First (* Jean-François Alcover, Feb 18 2014 *)
RealDigits[QPochhammer[1/2, 1/9], 10, 120][[1]] (* Amiram Eldar, May 08 2023 *)

Equals lim inf_{n->oo} Product_{k=0..floor(log_9(n))} floor(n/9^k)*9^k/n.
Equals lim inf_{n->oo} A132033(n)/n^(1+floor(log_9(n)))*9^(1/2*(1+floor(log_9(n)))*floor(log_9(n))).
Equals lim inf_{n->oo} A132033(n)/n^(1+floor(log_9(n)))*9^A000217(floor(log_9(n))).
Equals (1/2)*exp(-Sum_{n>0} 9^(-n)*Sum_{k|n} 1/(k*2^k)).
Equals lim inf_{n->oo} A132033(n)/A132033(n+1).
Equals Product_{n>=1} (1 - 1/A270369(n)). - Amiram Eldar, May 08 2023

A132021 Decimal expansion of Product_{k>=0} 1-1/(2*5^k).

Original entry on oeis.org

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

Offset: 0

Hieronymus Fischer, Aug 14 2007

			0.438796837203638531...

Cf. A048651, A098844, A067080, A132019, A132026, A132029, A100222, A000217, A020729.

digits = 103; NProduct[1-1/(2*5^k), {k, 0, Infinity}, NProductFactors -> 100, WorkingPrecision -> digits+5] // N[#, digits+5]& // RealDigits[#, 10, digits]& // First (* Jean-François Alcover, Feb 18 2014 *)
RealDigits[QPochhammer[1/2, 1/5], 10, 120][[1]] (* Amiram Eldar, May 08 2023 *)

Equals lim inf_{n->oo} Product_{k=0..floor(log_5(n))} floor(n/5^k)*5^k/n.
Equals lim inf_{n->oo} A132029(n)/n^(1+floor(log_5(n)))*5^(1/2*(1+floor(log_5(n)))*floor(log_5(n))).
Equals lim inf_{n->oo} A132029(n)/n^(1+floor(log_5(n)))*5^A000217(floor(log_5(n))).
Equals (1/2)*exp(-Sum_{n>0} 5^(-n)*Sum_{k|n} 1/(k*2^k)).
Equals lim inf_{n->oo} A132029(n)/A132029(n+1).
Equals Product_{n>=0} (1 - 1/A020729(n)). - Amiram Eldar, May 08 2023

A132031 Product{0<=k<=floor(log_7(n)), floor(n/7^k)}, n>=1.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 28, 30, 32, 34, 36, 38, 40, 63, 66, 69, 72, 75, 78, 81, 112, 116, 120, 124, 128, 132, 136, 175, 180, 185, 190, 195, 200, 205, 252, 258, 264, 270, 276, 282, 288, 343, 350, 357, 364, 371, 378, 385, 448, 456, 464, 472, 480, 488

Offset: 1

Hieronymus Fischer, Aug 20 2007

nonn

If n is written in base-7 as n=d(m)d(m-1)d(m-2)...d(2)d(1)d(0) (where d(k) is the digit at position k) then a(n) is also the product d(m)d(m-1)d(m-2)...d(2)d(1)d(0)*d(m)d(m-1)d(m-2)...d(2)d(1)*d(m)d(m-1)d(m-2)...d(2)*...*d(m)d(m-1)d(m-2)*d(m)d(m-1)*d(m).

			a(52)=floor(52/7^0)*floor(52/7^1)*floor(52/7^2)=52*7*1=364.
a(58)=464 since 58=112(base-7) and so a(58)=112*11*1(base-7)=58*8*1=464.

Cf. A048651, A132023, A132035, A000217.
For formulas regarding a general parameter p (i.e. terms floor(n/p^k)) see A132264.
For the product of terms floor(n/p^k) for p=2 to p=12 see A098844(p=2), A132027(p=3)-A132033(p=9), A067080(p=10), A132263(p=11), A132264(p=12).
For the products of terms 1+floor(n/p^k) see A132269-A132272, A132327, A132328.

Table[Times@@Floor[n/7^Range[0,Floor[Log[7,n]]]],{n,70}] (* Harvey P. Dale, Oct 11 2017 *)

Recurrence: a(n)=n*a(floor(n/7)); a(n*7^m)=n^m*7^(m(m+1)/2)*a(n).

a(k*7^m)=k^(m+1)*7^(m(m+1)/2), for 0

Asymptotic behavior: a(n)=O(n^((1+log_7(n))/2)this follows from the inequalities below.

a(n)<=b(n), where b(n)=n^(1+floor(log_7(n)))/7^((1+floor(log_7(n)))*floor(log_7(n))/2); equality holds for n=k*7^m, 0=0. b(n) can also be written n^(1+floor(log_7(n)))/7^A000217(floor(log_7(n))).

Also: a(n)<=3^((1-log_7(3))/2)*n^((1+log_7(n))/2)=1.270209197...*7^A000217(log_7(n)), equality holds for n=3*7^m, m>=0.

a(n)>c*b(n), where c=0.4587667266997689850200... (see constant A132023).

Also: a(n)>c*(sqrt(2)/2^log_7(sqrt(2)))*n^((1+log_7(n))/2)=0.4587667266...*1.249972544...*7^A000217(log_7(n)).

lim inf a(n)/b(n)=0.4587667266997689850200..., for n-->oo.

lim sup a(n)/b(n)=1, for n-->oo.

lim inf a(n)/n^((1+log_7(n))/2)=0.4587667266997689850200...*sqrt(2)/2^log_7(sqrt(2)), for n-->oo.

lim sup a(n)/n^((1+log_7(n))/2)=sqrt(3)/3^log_7(sqrt(3))=1.270209197..., for n-->oo.

lim inf a(n)/a(n+1)=0.4587667266997689850200... for n-->oo (see constant A132023).

A163467 a(n) = floor(p/2) * floor(floor(p/2)/2) * floor(floor(floor(p/2)/2)/2) * ... * 1, where p=prime(n).

Original entry on oeis.org

1, 1, 2, 3, 10, 18, 64, 72, 110, 294, 315, 1296, 2000, 2100, 2530, 6084, 8526, 9450, 33792, 38080, 46656, 53352, 82000, 106480, 248832, 270000, 275400, 322452, 341172, 460992, 615195, 2129920, 2515456, 2552448, 3548448, 3596400, 4161456

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jul 28 2009

Cumulative product of the residuals of a repeated shift-right operation on the base-2 representation of prime(n).

			For n=6, p=13, the intermediate factors are floor(13/2)=6, floor(6/2)=3, floor(3/2)=1, which yield a(6)=6*3*1=18.
For n=7, p=17, floor(17/2)=8, floor(8/2)=4, floor(4/2)=2, floor(2/2)=1, which yield a(7)=8*4*2*1=64.

G. C. Greubel, Table of n, a(n) for n = 1..5000

Cf. A098844.

lst={};Do[p=Prime[n];s=1;While[p>1,p=IntegerPart[p/2];s*=p;];AppendTo[lst,s],{n,5!}];lst
Table[Times@@Rest[NestWhileList[Floor[#/2]&,Prime[n],#>1&]],{n,40}] (* Harvey P. Dale, Jul 05 2019 *)

a(n) = my(p=prime(n), k=1); while(p!=1, k *= p\2; p = p\2); k; \\ Michel Marcus, Jul 26 2017

More divisions and primes mentioned in the definition by R. J. Mathar, Aug 02 2009

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A132023 Decimal expansion of Product_{k>=0} 1-1/(2*7^k).

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A132025 Decimal expansion of Product_{k>=0} 1-1/(2*9^k).

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A132021 Decimal expansion of Product_{k>=0} 1-1/(2*5^k).

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A132031 Product{0<=k<=floor(log_7(n)), floor(n/7^k)}, n>=1.

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A163467 a(n) = floor(p/2) * floor(floor(p/2)/2) * floor(floor(floor(p/2)/2)/2) * ... * 1, where p=prime(n).

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