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.

Previous Showing 11-15 of 15 results.

A015007 q-factorial numbers for q=8.

Original entry on oeis.org

1, 1, 9, 657, 384345, 1799118945, 67375205371305, 20185139902805378865, 48378633136349277767794425, 927610024989668734297857360967425, 142287668466497494704440569679875994730825, 174605966461872393482359052970987514818406771638225
Offset: 0

Views

Author

Olivier Gérard

Keywords

Links

Crossrefs

Cf. A015001, A015002, A015004, A015005, A015006, A015008, A015009, A015011.
Column q=8 of A069777.
Cf. A023001, A132036.

Programs

  • Magma
    [n le 1 select 1 else (8^n-1)*Self(n-1)/7: n in [1..15]]; // Vincenzo Librandi, Oct 26 2012
  • Mathematica
    RecurrenceTable[{a[1]==1, a[n]==((8^n - 1) * a[n-1])/7}, a, {n, 15}] (* Vincenzo Librandi, Oct 26 2012 *)
Table[QFactorial[n, 8], {n, 15}] (* Bruno Berselli, Aug 14 2013 *)

Formula

a(n) = Product_{k=1..n} ((q^k - 1) / (q - 1)), with q=8.
a(0) = 1, a(n) = (8^n-1)*a(n-1)/7. - Vincenzo Librandi, Oct 26 2012
From Amiram Eldar, Jul 05 2025: (Start)
a(n) = Product_{k=1..n} A023001(k).
a(n) ~ c * 8^(n*(n+1)/2)/7^n, where c = A132036. (End)

Extensions

a(0)=1 prepended by Alois P. Heinz, Sep 08 2021

A052496 Number of nonsingular n X n matrices over GF(8).

Original entry on oeis.org

1, 7, 3528, 115379712, 241909719367680, 32467582052437076213760, 278893342293098904613804037898240, 153323163270070838469523866093442017326530560
Offset: 0

Views

Author

Vladeta Jovovic, Mar 16 2000

Keywords

Links

Crossrefs

Cf. A027876, A109966.
Cf. A002884, A003791, A053290, A053291, A053292, A053293, A052497, A052498, A132036.

Programs

  • Magma
    [1] cat [&*[(8^n-8^k): k in [0..n-1]]: n in [1..10]]; // Bruno Berselli, Jan 30 2013
  • Mathematica
    Table[Product[(8^n - 8^j), {j, 0, n-1}], {n, 0, 10}] (* G. C. Greubel, May 14 2019 *)
  • PARI
    {a(n) = prod(j=0,n-1, 8^n - 8^j)}; \\ G. C. Greubel, May 14 2019
  • Sage
    [product(8^n - 8^j for j in (0..n-1)) for n in (0..10)] # G. C. Greubel, May 14 2019

Formula

a(n) = (8^n - 1)*(8^n - 8)*...*(8^n - 8^(n-1)).
a(n) = A109966(n)*A027876(n). - Bruno Berselli, Jan 30 2013
a(n) ~ c * 8^(n^2), where c = A132036. - Amiram Eldar, Jul 06 2025

A003791 Order of universal Chevalley group A_n (8).

Original entry on oeis.org

1, 504, 16482816, 34558531338240, 4638226007491010887680, 39841906041871272087686291128320, 21903309038581548352789123727634573903790080
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

References

  • J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker and R. A. Wilson, ATLAS of Finite Groups. Oxford Univ. Press, 1985 [for best online version see https://oeis.org/wiki/Welcome#Links_to_Other_Sites], p. xvi.
  • H. S. M. Coxeter and W. O. J. Moser, Generators and Relations for Discrete Groups, 4th ed., Springer-Verlag, NY, reprinted 1984, p. 131.

Links

Crossrefs

Cf. A003787, A003788, A003789, A003790, A003792, A052496, A132036.

Programs

  • Magma
    [&*[(8^n - 8^k): k in [0..n-1]]/7: n in [1..8]]; // Vincenzo Librandi, Sep 19 2015
  • Mathematica
    f[m_, n_] := m^(n (n + 1)/2) Product[m^k - 1, {k, 2, n + 1}]; f[8, #] & /@ Range[0, 6] (* Michael De Vlieger, Sep 18 2015 *)

Formula

Numbers so far appear to equal A052496(n)/7. - Ralf Stephan, Mar 30 2004
a(n) = A(8,n) where A(q,n) is defined in A003787. - Sean A. Irvine, Sep 18 2015
a(n) ~ c * 8^(n*(n+2)), where c = (8/7) * A132036 = 0.982178279315... . - Amiram Eldar, Jul 07 2025

A132024 Decimal expansion of Product_{k>=0} (1-1/(2*8^k)).

Original entry on oeis.org

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

Views

Author

Hieronymus Fischer, Aug 14 2007

Keywords

Examples

			0.46456888368647639098...

Crossrefs

Cf. A048651, A098844, A067080, A132019, A132026, A132032, A132036, A000217, A013730.

Programs

  • Mathematica
    RealDigits[QPochhammer[1/2,1/8],10,120][[1]] (* Harvey P. Dale, May 23 2011 *)
  • PARI
    prodinf(k=0, 1 - 1/(2*8^k)) \\ Amiram Eldar, May 09 2023

Formula

Equals lim inf_{n->oo} Product_{k=0..floor(log_8(n))} floor(n/8^k)*8^k/n.
Equals lim inf_{n->oo} A132032(n)/n^(1+floor(log_8(n)))*8^(1/2*(1+floor(log_8(n)))*floor(log_8(n))).
Equals lim inf_{n->oo} A132032(n)/n^(1+floor(log_8(n)))*8^A000217(floor(log_8(n))).
Equals (1/2)*exp(-Sum_{n>0} 8^(-n)*Sum_{k|n} 1/(k*2^k)).
Equals lim inf_{n->oo} A132032(n)/A132032(n+1).
Equals Product_{n>=0} (1 - 1/A013730(n)). - Amiram Eldar, May 09 2023

Extensions

Name corrected by Amiram Eldar, May 09 2023

A132032 Product{0<=k<=floor(log_8(n)), floor(n/8^k)}, n>=1.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 32, 34, 36, 38, 40, 42, 44, 46, 72, 75, 78, 81, 84, 87, 90, 93, 128, 132, 136, 140, 144, 148, 152, 156, 200, 205, 210, 215, 220, 225, 230, 235, 288, 294, 300, 306, 312, 318, 324, 330, 392, 399, 406, 413, 420, 427, 434
Offset: 1

Views

Author

Hieronymus Fischer, Aug 20 2007

Keywords

Comments

If n is written in base-8 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(70)=floor(70/8^0)*floor(70/8^1)*floor(70/8^2)=70*8*1=560;
For n=75, 75=113(base-8) and so a(75)=113*11*1(base-8)=75*9*1=675.

Crossrefs

Cf. A048651, A132024, A132036, 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.

Formula

Recurrence: a(n)=n*a(floor(n/8)); a(n*8^m)=n^m*8^(m(m+1)/2)*a(n).
a(k*8^m)=k^(m+1)*8^(m(m+1)/2), for 0
Asymptotic behavior: a(n)=O(n^((1+log_8(n))/2)); this follows from the inequalities below.
a(n)<=b(n), where b(n)=n^(1+floor(log_8(n)))/8^((1+floor(log_8(n)))*floor(log_8(n))/2); equality holds for n=k*8^m, 0=0. b(n) can also be written n^(1+floor(log_8(n)))/8^A000217(floor(log_8(n))).
Also: a(n)<=3^((1-log_8(3))/2)*n^((1+log_8(n))/2) = 1.295758534...*8^A000217(log_8(n)), equality holds for n=3*8^m, m>=0.
a(n)>c*b(n), where c = 0.46456888368647639098... (see constant A132024).
Also: a(n)>c*2^(1/3)*n^((1+log_8(n))/2)=0.4645688836...*1.25992105...*8^A000217(log_8(n)).
lim inf a(n)/b(n)=0.46456888368647639098..., for n-->oo.
lim sup a(n)/b(n)=1, for n-->oo.
lim inf a(n)/n^((1+log_8(n))/2)=0.46456888368647639098...*2^(1/3), for n-->oo.
lim sup a(n)/n^((1+log_8(n))/2)=sqrt(3)/3^log_8(sqrt(3))=1.295758534..., for n-->oo.
lim inf a(n)/a(n+1)=0.46456888368647639098... for n-->oo (see constant A132024).
Previous Showing 11-15 of 15 results.

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

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