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 1-6 of 6 results.

A133304 Number of distinct prime factors of A101291.

Original entry on oeis.org

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

Views

Author

Parthasarathy Nambi, Oct 18 2007

Keywords

Examples

			The number of distinct prime factors of 45 is 2.
The number of distinct prime factors of 4905 is 3.
The number of distinct prime factors of 494550 is 5.

Links

Crossrefs

Cf. A101291.

Programs

  • Maple
    A101291 := proc(n) 99*100^n/200-9*10^n/20 ; end: A133304 := proc(n) nops(numtheory[factorset](A101291(n))) ; end: for n from 1 do printf("%d,\n",A133304(n)) ; od: # R. J. Mathar, Jul 08 2009
  • Mathematica
    f[n_] := 10^n(10^n - 1)/2; Table[PrimeNu[f[n] - f[n - 1]], {n, 60}] (* James C. McMahon, Mar 14 2025 *)

Extensions

28 more terms from R. J. Mathar, Jul 08 2009
a(41)-a(60) from James C. McMahon, Mar 14 2025

A010036 Sum of 2^n, ..., 2^(n+1) - 1.

Original entry on oeis.org

1, 5, 22, 92, 376, 1520, 6112, 24512, 98176, 392960, 1572352, 6290432, 25163776, 100659200, 402644992, 1610596352, 6442418176, 25769738240, 103079084032, 412316598272, 1649266917376, 6597068718080, 26388276969472, 105553112072192, 422212456677376
Offset: 0

Views

Author

Steve King (ITTTUCSON(AT)aol.com)

Keywords

Comments

a(n) = sum of next 2^n natural numbers. - Amarnath Murthy, Apr 17 2003
Sum of all proper binary numbers with n digits (i.e. those not beginning with 0). Cf. A101291 Sum of all numbers with n digits [base 10]. - Jonathan Vos Post, Sep 07 2006
a(n)/2^n gives the average eccentricity of the graphs of the Chinese rings puzzle with n+1 rings (also known as baguenaudier). - Daniele Parisse, Jun 02 2008

Links

Crossrefs

Cf. A010036.
Partial sums are in A006516, A006095.

Programs

  • Magma
    [ &+[ k: k in [2^n..2^(n+1)-1] ]: n in [0..21] ]; // Klaus Brockhaus, Nov 27 2009
  • Magma
    [2^n *(2^n+(2^(n+1)-1))/2: n in [0..25]]; // Vincenzo Librandi, Sep 11 2015
  • Maple
    f:= n-> 3*2^(2*n-1)-2^(n-1): seq(f(n), n=0..30);
  • Mathematica
    Table[2^n (2^n+(2^(n+1)-1))/2,{n,0,25}] (* or *) LinearRecurrence[{6,-8},{1,5},30] (* Harvey P. Dale, Jan 23 2012 *)
With[{nn=30},Total/@TakeList[Range[2^(nn+1)-1],2^Range[0,nn]]] (* Harvey P. Dale, May 26 2024 *)
  • PARI
    a(n)=3<<(2*n-1)-1<<(n-1) \\ Charles R Greathouse IV, Jul 02 2013

Formula

a(n+1) = 4*a(n) + 2^n with a(0) = 1 (with a(0)=0, see A006516). a(n) = 2^(n-1)*A055010(n). - Philippe Deléham, Feb 20 2004
a(n) = 3*2^(2*n-1) - 2^(n-1). - Daniele Parisse, Jun 10 2007
From Klaus Brockhaus, Nov 27 2009: (Start)
a(n) = 6*a(n-1)-8*a(n-2) for n > 1; a(0) = 1, a(1) = 5.
G.f.: (1-x)/((1-2*x)*(1-4*x)). (End)
a(n) = Sum_{k, 0<=k<=n} A125185(n,k)*2^k. - Philippe Deléham, Feb 26 2012
a(n) = A006516(n+1)-A006516(n). - R. J. Mathar, Mar 06 2017

A037182 a(n) = 10^n*(10^n-1) / 2.

Original entry on oeis.org

0, 45, 4950, 499500, 49995000, 4999950000, 499999500000, 49999995000000, 4999999950000000, 499999999500000000, 49999999995000000000, 4999999999950000000000, 499999999999500000000000, 49999999999995000000000000, 4999999999999950000000000000
Offset: 0

Views

Author

Marvin Ray Burns

Keywords

Comments

Sum of all numbers with <= n digits.

Links

Crossrefs

Partial sums of A101291.

Programs

  • Mathematica
    a[n_] := 10^n(10^n - 1)/2; Table[ a[n], {n, 0, 12}] (* Robert G. Wilson v, Dec 24 2004 *)

Formula

Sum_{i=1..10^n-1} i = the (10^n-1)th triangular number (A000217). - Marvin Ray Burns
From Chai Wah Wu, Mar 17 2018: (Start)
a(n) = 110*a(n-1) - 1000*a(n-2) for n > 1.
G.f.: 45*x/((10*x - 1)*(100*x - 1)). (End)

A121544 Sum of all proper base 4 numbers with n digits (those not beginning with 0).

Original entry on oeis.org

6, 114, 1896, 30624, 491136, 7862784, 125822976, 2013241344, 32212156416, 515395682304, 8246335635456, 131941389041664, 2111062300164096, 33776997104615424, 540431954881806336, 8646911282940739584, 138350580546379186176, 2213609288819376390144
Offset: 1

Views

Author

Jonathan Vos Post, Sep 08 2006

Keywords

Comments

Sum of the first 3 * 4^(n-1) integers starting with 4^(n-1).
Sum of the integers from 4^(n-1) to 4^n -1.
First differences of A026337.

Examples

			a(1) = 6 = 1 + 2 + 3.
a(2) = 114 = 10_4 + 11_4 + 12_4 + 13_4 + 20_4 + 21_4 + 22_4 + 23_4 + 30_4 + 31_4 + 32_4 + 33_4 = (4+5+6+7+8+9+10+11+12+13+14+15)_10.

Links

Crossrefs

Cf. A007090, A010036, A026121, A026337, A101291.

Programs

  • Magma
    [3*Binomial(5*4^(n-1), 2)/5: n in [1..20]]; // G. C. Greubel, Nov 07 2024
  • Mathematica
    Table[3*4^(n-1)*(5*4^(n-1) - 1)/2, {n,20}] (* James C. McMahon, Oct 19 2024 *)
  • SageMath
    def A121544(n): return 3*binomial(5*4^(n-1), 2)//5
[A121544(n) for n in range(1,21)] # G. C. Greubel, Nov 07 2024

Formula

a(n) = 3 * 4^(n-1) * (4^(n-1) + 4^n - 1)/2.
G.f.: 6*x*(1-x) / ((1-4*x)*(1-16*x)). - Colin Barker, Apr 30 2013
From G. C. Greubel, Nov 07 2024: (Start)
a(n) = (3/5)*binomial(5*4^(n-1), 2).
E.g.f.: (3/32)*(-1 - 4*exp(4*x) + 5*exp(16*x)). (End)

Extensions

More terms from Colin Barker, Apr 30 2013
Edited by Michel Marcus, Apr 15 2024

A226508 a(n) = Sum_{i=3^n..3^(n+1)-1} i.

Original entry on oeis.org

3, 33, 315, 2889, 26163, 235953, 2125035, 19129689, 172180323, 1549662273, 13947078555, 125524061289, 1129717614483, 10167461718993, 91507165036875, 823564514029689, 7412080712360643, 66708726669526113, 600378540800575995, 5403406869529706889
Offset: 0

Views

Author

Michel Marcus, Jun 10 2013

Keywords

Comments

Partial sums give 3, 36, 351, 3240, 29403,...: A026121.
a(n) is the sum of all integers having n+1 digits in their ternary expansion (without leading zeros). - Jonathan Vos Post, Sep 07 2006

Examples

			a(0) = 1+2 = 3,
a(1) = 3+4+5+6+7+8 = 33.

Links

Crossrefs

Cf. A010035, A010036 (base 2), A026121, A101291 (base 10).
Cf. A007089 (numbers in base 3).

Programs

  • Mathematica
    Table[3^(n - 1) (4 3^(n + 1) - 3), {n, 0, 20}] (* Bruno Berselli, Jun 11 2013 *)
LinearRecurrence[{12,-27},{3,33},30] (* Harvey P. Dale, Jun 19 2013 *)
  • PARI
    a(n) = sum(i=3^n, 3^(n+1)-1, i) \\ Michel Marcus, Jun 11 2013

Formula

G.f.: 3*(1-x)/(1-12*x+27*x^2). [Bruno Berselli, Jun 11 2013]
a(n) = 3^(n-1)*(4*3^(n+1)-3). [Bruno Berselli, Jun 11 2013]
a(0)=3, a(1)=33, a(n)=12*a(n-1)-27*a(n-2). - Harvey P. Dale, Jun 19 2013

A291658 a(n) is the sum of all integers from 5^n to 5^(n+1)-1.

Original entry on oeis.org

10, 290, 7450, 187250, 4686250, 117181250, 2929656250, 73242031250, 1831053906250, 45776363281250, 1144409160156250, 28610229394531250, 715255736816406250, 17881393430175781250, 447034835803222656250, 11175870895324707031250, 279396772384338378906250
Offset: 0

Views

Author

Enrique Navarrete, Aug 28 2017

Keywords

Comments

a(n) is the sum of all (positive) numbers having exactly (n+1) digits when written in base 5. - Alois P. Heinz, Sep 25 2017

Examples

			For n=0, the sum is from 1 to 4, so a(0)=10;
for n=1, the sum is from 5 to 24, so a(1)=290, etc.

Links

Crossrefs

Cf. A010036, A226508, A121544, A101291, A162729.

Programs

  • Maple
    a:= unapply(sum(i, i=5^n..5^(n+1)-1), n):
seq(a(n), n=0..20);  # Alois P. Heinz, Sep 25 2017
  • PARI
    Vec(10*(1 - x) / ((1 - 5*x)*(1 - 25*x)) + O(x^30)) \\ Colin Barker, Sep 12 2017

Formula

a(n) = ((5^n)/2)*(5^(n+2) - 5^n - 4), n >= 0.
From Colin Barker, Sep 12 2017: (Start)
G.f.: 10*(1 - x) / ((1 - 5*x)*(1 - 25*x)).
a(n) = 30*a(n-1) - 125*a(n-2) for n>1.
(End)
a(n) = A162729(n+1) - A162729(n). - Alois P. Heinz, Sep 25 2017
Showing 1-6 of 6 results.

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