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 11-18 of 18 results.

A028667 Galois numbers for p=5; order of group AGL(n,5).

Original entry on oeis.org

1, 20, 12000, 186000000, 72540000000000, 708171750000000000000, 172882428468750000000000000000, 1055177097007236328125000000000000000000, 161006835289591673217773437500000000000000000000000, 614192019859664935862872123718261718750000000000000000000000000
Offset: 0

Views

Author

Olivier Gérard

Keywords

Links

Crossrefs

Cf. A028365, A028665, A028669, A028673, A028675, A028679, A028681, A028685.
Cf. A100222.

Programs

  • Mathematica
    FoldList[ #1*5^#2 (5^#2-1)&, 1, Range[ 20 ] ]
a[n_] := 5^n * Product[5^n - 5^k, {k, 0, n-1}]; Array[a, 10, 0] (* Amiram Eldar, Jul 12 2025 *)
  • PARI
    a(n) = 5^n * prod(k = 0, n-1, 5^n - 5^k); \\ Amiram Eldar, Jul 12 2025

Formula

a(n) = 5^n * Product_{k=0..n-1} (5^n - 5^k).
a(n) ~ c * 5^(n^2+n), where c = A100222. - Amiram Eldar, Jul 12 2025

A003789 Order of universal Chevalley group A_n (5).

Original entry on oeis.org

1, 120, 372000, 29016000000, 56653740000000000, 2766118855500000000000000, 3376566710423156250000000000000000, 103044374585338670859375000000000000000000000
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, A003790, A003791, A003792, A053292, A100222.

Programs

  • Magma
    [&*[(5^n-5^k): k in [0..n-1]]/4: 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[5, #] & /@ Range[0, 7] (* Michael De Vlieger, Sep 18 2015 *)

Formula

Numbers so far appear to equal A053292(n)/4. - Ralf Stephan, Mar 30 2004
a(n) = A(5,n) where A(q,n) is defined in A003787. - Sean A. Irvine, Sep 18 2015
a(n) ~ c * 5^(n*(n+2)), where c = (5/4) * A100222 = 0.950415994839... . - Amiram Eldar, Jul 07 2025

A132029 Product{0<=k<=floor(log_5(n)), floor(n/5^k)}, n>=1.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 20, 22, 24, 26, 28, 45, 48, 51, 54, 57, 80, 84, 88, 92, 96, 125, 130, 135, 140, 145, 180, 186, 192, 198, 204, 245, 252, 259, 266, 273, 320, 328, 336, 344, 352, 405, 414, 423, 432, 441, 1000, 1020, 1040, 1060, 1080, 1210, 1232, 1254, 1276
Offset: 1

Views

Author

Hieronymus Fischer, Aug 20 2007

Keywords

Comments

If n is written in base-5 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(26)=floor(26/5^0)*floor(26/5^1)*floor(26/5^2)=26*5*1=130; a(34)=204 since 34=114(base-5) and so a(34)=114*11*1(base-5)=34*6*1=204.

Links

Crossrefs

Cf. A048651, A132021, A100222, 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[Product[Floor[n/5^k],{k,0,Floor[Log[5,n]]}],{n,60}] (* Harvey P. Dale, Oct 16 2019 *)

Formula

Recurrence: a(n)=n*a(floor(n/5)); a(n*5^m)=n^m*5^(m(m+1)/2)*a(n).
a(k*5^m)=k^(m+1)*5^(m(m+1)/2), for 0
Asymptotic behavior: a(n)=O(n^((1+log_5(n))/2)); this follows from the inequalities below.
a(n)<=b(n), where b(n)=n^(1+floor(log_5(n)))/5^((1+floor(log_5(n)))*floor(log_5(n))/2); equality holds for n=k*5^m, 0=0. b(n) can also be written n^(1+floor(log_5(n)))/5^A000217(floor(log_5(n))).
Also: a(n)<=2^((1-log_5(2))/2)*n^((1+log_5(n))/2)=1.2181246...*5^A000217(log_5(n)), equality holds for n=2*5^m, m>=0.
a(n)>c*b(n), where c=0.438796837203638531... (see constant A132021).
Also: a(n)>c*(sqrt(2)/2^log_5(sqrt(2)))*n^((1+log_5(n))/2)=0.534509224...*5^A000217(log_5(n)).
lim inf a(n)/b(n)=0.438796837203638531..., for n-->oo.
lim sup a(n)/b(n)=1, for n-->oo.
lim inf a(n)/n^((1+log_5(n))/2)=0.438796837203638531...*sqrt(2)/2^log_5(sqrt(2)), for n-->oo.
lim sup a(n)/n^((1+log_5(n))/2)=sqrt(2)/2^log_5(sqrt(2))=1.2181246..., for n-->oo.
lim inf a(n)/a(n+1)=0.438796837203638531... for n-->oo (see constant A132021).

A258460 Number of partitions of n into parts of exactly 5 sorts which are introduced in ascending order.

Original entry on oeis.org

1, 16, 157, 1223, 8331, 52078, 307122, 1738441, 9552809, 51357781, 271624053, 1418856775, 7341440755, 37708531955, 192586153199, 979219591861, 4961598056587, 25071026497266, 126410385360189, 636282269208285, 3198360708483673, 16059685003763157
Offset: 5

Views

Author

Alois P. Heinz, May 30 2015

Keywords

Links

Crossrefs

Column k=5 of A256130.
Cf. A320547.

Programs

  • Maple
    b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0,
      b(n, i-1, k) +`if`(i>n, 0, k*b(n-i, i, k))))
    end:
T:= (n, k)-> add(b(n$2, k-i)*(-1)^i/(i!*(k-i)!), i=0..k):
a:= n-> T(n,5):
seq(a(n), n=5..30);
  • Mathematica
    b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i < 1, 0, b[n, i - 1, k] + If[i > n, 0, k*b[n - i, i, k]]]];
T[n_, k_] := Sum[b[n, n, k - i]*(-1)^i/(i!*(k - i)!), {i, 0, k}];
a[n_] := T[n, 5];
Table[a[n], {n, 5, 30}] (* Jean-François Alcover, May 22 2018, translated from Maple *)

Formula

a(n) ~ c * 5^n, where c = 1/(5!*Product_{n>=1} (1-1/5^n)) = 1/(5!*QPochhammer[1/5, 1/5]) = 1/(5!*A100222) = 0.0109601129644612101609007882... . - Vaclav Kotesovec, Jun 01 2015

A371752 Decimal expansion of Product_{k>=0} 1 / (1 + 1/5^k).

Original entry on oeis.org

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

Views

Author

Ilya Gutkovskiy, Apr 05 2024

Keywords

Examples

			0.39666773502905896079231190779208218756...

Crossrefs

Cf. A000351, A083864, A100222, A371746, A371749, A371750, A371751.

Programs

  • Mathematica
    RealDigits[1/QPochhammer[-1, 1/5], 10, 100][[1]]

A371750 Decimal expansion of Product_{k>=1} 1 / (1 - 1/5^k).

Original entry on oeis.org

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

Views

Author

Ilya Gutkovskiy, Apr 05 2024

Keywords

Examples

			1.3152135557353452193080945851009617844...

Links

Crossrefs

Cf. A000351, A065446, A100222, A370466, A371747, A371751, A371752.

Programs

  • Mathematica
    RealDigits[1/QPochhammer[1/5, 1/5], 10, 100][[1]]

Formula

Equals 1 / A100222.

A371751 Decimal expansion of Product_{k>=0} (1 + 1/5^k).

Original entry on oeis.org

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

Views

Author

Ilya Gutkovskiy, Apr 05 2024

Keywords

Examples

			2.5210016134202150647777463154782130132...

Links

Crossrefs

Cf. A000351, A081845, A100222, A132323, A371748, A371750, A371752.

Programs

  • Mathematica
    RealDigits[QPochhammer[-1, 1/5], 10, 100][[1]]
RealDigits[Times@@Table[1+1/5^k,{k,0,1000}],10,100][[1]] (* Harvey P. Dale, Sep 17 2024 *)

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

Views

Author

Hieronymus Fischer, Aug 14 2007

Keywords

Examples

			0.438796837203638531...

Crossrefs

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

Programs

  • Mathematica
    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 *)

Formula

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
Previous Showing 11-18 of 18 results.

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

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