cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A132035 Decimal expansion of Product_{k>0} (1-1/7^k).

Original entry on oeis.org

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

Views

Author

Hieronymus Fischer, Aug 14 2007

Keywords

Examples

			0.8367954070890378710...

Links

Crossrefs

Cf. A027875, A048651, A100220, A132023, A132026, A132038, A098844, A067080, A000203.

Programs

  • Mathematica
    digits = 103; NProduct[1-1/7^k, {k, 1, Infinity}, NProductFactors -> 200, WorkingPrecision -> digits+3] // N[#, digits+3]& // RealDigits[#, 10, digits]& // First (* Jean-François Alcover, Feb 18 2014 *)
RealDigits[QPochhammer[1/7], 10, 100][[1]] (* Amiram Eldar, May 09 2023 *)
  • PARI
    prodinf(k=1, 1 - 1/(7^k)) \\ Amiram Eldar, May 09 2023

Formula

Equals exp(-Sum_{n>0} sigma_1(n)/(n*7^n)) = exp(-Sum_{n>0} A000203(n)/(n*7^n)).
From Amiram Eldar, May 09 2023: (Start)
Equals sqrt(2*Pi/log(7)) * exp(log(7)/24 - Pi^2/(6*log(7))) * Product_{k>=1} (1 - exp(-4*k*Pi^2/log(7))) (McIntosh, 1995).
Equals Sum_{n>=0} (-1)^n/A027875(n). (End)

A109808 a(n) = 2*7^(n-1).

Original entry on oeis.org

2, 14, 98, 686, 4802, 33614, 235298, 1647086, 11529602, 80707214, 564950498, 3954653486, 27682574402, 193778020814, 1356446145698, 9495123019886, 66465861139202, 465261027974414, 3256827195820898, 22797790370746286, 159584532595224002, 1117091728166568014
Offset: 1

Views

Author

Woong Kook (andrewk(AT)math.uri.edu), Aug 16 2005

Keywords

Comments

Value of Tutte dichromatic polynomial T_G(0,1) where G is the Cartesian product of the paths P_2 and P_n (n>1).
The value of Tutte dichromatic polynomial T_G(0,1) where G is the Cartesian product of the paths P_1 and P_n (n>1) is seen to be 2^(n-1), which is also the number of edge-rooted forests in P_n.
In 1956, Andrzej Schinzel showed that for every n >= 2, a(n) is not a value of Euler's function. - Arkadiusz Wesolowski, Oct 20 2013
Apart from first term 2, these are the numbers that satisfy phi(n) = 3*n/7. - Michel Marcus, Jul 14 2015

Links

Crossrefs

Cf. A000420 (powers of 7), A005277 (nontotients), A132023.

Programs

Formula

a(n) = 2*7^(n-1); a(n) = 7*a(n-1) where a(1) = 2.
G.f.: 2*x/(1 - 7*x). - Philippe Deléham, Nov 23 2008
E.g.f.: 2*(exp(7*x) - 1)/7. - Stefano Spezia, May 29 2021
From Amiram Eldar, May 08 2023: (Start)
Sum_{n>=1} 1/a(n) = 7/12.
Sum_{n>=1} (-1)^(n+1)/a(n) = 7/16.
Product_{n>=1} (1 - 1/a(n)) = A132023. (End)

Extensions

Name changed by Arkadiusz Wesolowski, Oct 20 2013

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

Views

Author

Hieronymus Fischer, Aug 20 2007

Keywords

Comments

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).

Examples

			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.

Crossrefs

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.

Programs

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

Formula

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).
Showing 1-3 of 3 results.

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