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.

A132037 Decimal expansion of Product_{k>0} (1-1/9^k).

Original entry on oeis.org

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

Views

Author

Hieronymus Fischer, Aug 14 2007

Keywords

Examples

			0.8765603540359642058360198...

Links

Crossrefs

Cf. A010815, A027877, A048651, A100220, A132025, A132026, A132038, A098844, A067080, A000203.

Programs

  • Mathematica
    digits = 103; NProduct[1-1/9^k, {k, 1, Infinity}, NProductFactors -> 100, WorkingPrecision -> digits+3] // N[#, digits+3]& // RealDigits[#, 10, digits]& // First (* Jean-François Alcover, Feb 18 2014 *)
RealDigits[QPochhammer[1/9], 10, 100][[1]] (* Amiram Eldar, May 09 2023 *)
  • PARI
    prodinf(k=1, 1 - 1/(9^k)) \\ Amiram Eldar, May 09 2023

Formula

Equals exp(-Sum_{n>0} sigma_1(n)/(n*9^n)) = exp(-Sum_{n>0} A000203(n)/(n*9^n)).
Equals Sum_{n>=0} A010815(n)/9^n. - R. J. Mathar, Mar 04 2009
From Amiram Eldar, May 09 2023: (Start)
Equals sqrt(Pi/log(3)) * exp(log(3)/12 - Pi^2/(12*log(3))) * Product_{k>=1} (1 - exp(-2*k*Pi^2/log(3))) (McIntosh, 1995).
Equals Sum_{n>=0} (-1)^n/A027877(n). (End)

A132033 Product{0<=k<=floor(log_9(n)), floor(n/9^k)}, n>=1.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 36, 38, 40, 42, 44, 46, 48, 50, 52, 81, 84, 87, 90, 93, 96, 99, 102, 105, 144, 148, 152, 156, 160, 164, 168, 172, 176, 225, 230, 235, 240, 245, 250, 255, 260, 265, 324, 330, 336, 342, 348, 354, 360, 366, 372
Offset: 1

Views

Author

Hieronymus Fischer, Aug 20 2007

Keywords

Comments

If n is written in base-9 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(85)=floor(85/9^0)*floor(85/9^1)*floor(85/9^2)=85*9*1=765; a(88)=792 since 88=107(base-9) and so a(88)=107*10*1(base-9)=88*9*1=792.

Crossrefs

Cf. A048651, A132025, A132037, 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)-A132032(p=8), 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[Product[Floor[n/9^k],{k,0,Floor[Log[9,n]]}],{n,62}] (* James C. McMahon, Mar 03 2025 *)

Formula

Recurrence: a(n)=n*a(floor(n/9)); a(n*9^m)=n^m*9^(m(m+1)/2)*a(n).
a(k*9^m)=k^(m+1)*9^(m(m+1)/2), for 0
Asymptotic behavior: a(n)=O(n^((1+log_9(n))/2)); this follows from the inequalities below.
a(n)<=b(n), where b(n)=n^(1+floor(log_9(n)))/9^((1+floor(log_9(n)))*floor(log_9(n))/2); equality holds for n=k*9^m, 0=0. b(n) can also be written n^(1+floor(log_9(n)))/9^A000217(floor(log_9(n))).
Also: a(n)<=3^(1/4)*n^((1+log_9(n))/2)=1.316074013...*9^A000217(log_9(n)), equality holds for n=3*9^m, m>=0.
a(n)>c*b(n), where c=0.4689451783670236932832800... (see constant A132024).
Also: a(n)>c*2^((1-log_9(2))/2)*n^((1+log_9(n))/2)=0.4689451783670...*1.267747616...*9^A000217(log_9(n)).
lim inf a(n)/b(n)=0.4689451783670236932832800..., for n-->oo.
lim sup a(n)/b(n)=1, for n-->oo.
lim inf a(n)/n^((1+log_9(n))/2)=0.4689451783670236932832800...*sqrt(2)/2^log_9(sqrt(2)), for n-->oo.
lim sup a(n)/n^((1+log_9(n))/2)=3^(1/4)=1.316074013..., for n-->oo.
lim inf a(n)/a(n+1)=0.4689451783670236932832800... for n-->oo (see constant A132025).

A270369 Expansion of g.f. (1-7*x)/(1-9*x).

Original entry on oeis.org

1, 2, 18, 162, 1458, 13122, 118098, 1062882, 9565938, 86093442, 774840978, 6973568802, 62762119218, 564859072962, 5083731656658, 45753584909922, 411782264189298, 3706040377703682, 33354363399333138, 300189270593998242, 2701703435345984178, 24315330918113857602, 218837978263024718418
Offset: 0

Views

Author

Colin Barker, Mar 18 2016

Keywords

Links

Crossrefs

Cf. A001019 (powers of 9), A054879 (partial sums), A132025.
Cf. similar sequences with g.f. (1-k*x)/(1-9*x) and k=0..8: A001019 (k=0; k=8 gives two initial 1's ), A055275 (k=1), A270472 (k=2), A092810 (k=3), A067403 (k=4), A270473 (k=5), A102518 (k=6), this sequence (k=7).

Programs

  • Mathematica
    CoefficientList[Series[(1-7x)/(1-9x),{x,0,20}],x] (* or *) Join[ {1}, NestList[9#&,2,20]] (* Harvey P. Dale, Oct 15 2017 *)
  • PARI
    Vec((1-7*x)/(1-9*x) + O(x^30))

Formula

G.f.: (1-7*x)/(1-9*x).
a(n) = 9*a(n-1) for n>1.
a(n) = 2*9^(n-1) for n>0.
From Amiram Eldar, May 08 2023: (Start)
Sum_{n>=0} 1/a(n) = 25/16.
Sum_{n>=0} (-1)^n/a(n) = 11/20.
Product_{n>=1} (1 - 1/a(n)) = A132025. (End)
E.g.f.: (2*exp(9*x) + 7)/9. - Elmo R. Oliveira, Mar 25 2025
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.