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 1-10 of 11 results. Next

A161713 a(n) = (-n^5 + 15*n^4 - 65*n^3 + 125*n^2 - 34*n + 40)/40.

Original entry on oeis.org

1, 2, 4, 7, 14, 28, 49, 71, 79, 46, -70, -329, -812, -1624, -2897, -4793, -7507, -11270, -16352, -23065, -31766, -42860, -56803, -74105, -95333, -121114, -152138, -189161, -233008, -284576, -344837, -414841, -495719, -588686, -695044
Offset: 0

Views

Author

Reinhard Zumkeller, Jun 17 2009

Keywords

Comments

{a(k): 0 <= k < 6} = divisors of 28:
a(n) = A027750(A006218(27) + k + 1), 0 <= k < A000005(28).

Examples

			Differences of divisors of 28 to compute the coefficients of their interpolating polynomial, see formula:
  1     2     4     7    14    28
     1     2     3     7    14
        1     1     4     7
           0     3     3
              3     0
                -3
		

Crossrefs

Programs

  • Magma
    [(-n^5 + 15*n^4 - 65*n^3 + 125*n^2 - 34*n + 40)/40: n in [0..40]]; // Vincenzo Librandi, Jul 17 2011
    
  • Mathematica
    Table[(-n^5+15n^4-65n^3+125n^2-34n)/40+1,{n,0,40}] (* or *) LinearRecurrence[{6,-15,20,-15,6,-1},{1,2,4,7,14,28},40] (* Harvey P. Dale, Jan 14 2014 *)
  • PARI
    a(n)=(-n^5+15*n^4-65*n^3+125*n^2-34*n+40)/40 \\ Charles R Greathouse IV, Sep 24 2015
    
  • Python
    def A161713(n): return n*(n*(n*(n*(15 - n) - 65) + 125) - 34)//40 + 1 # Chai Wah Wu, Dec 16 2021

Formula

a(n) = C(n,0) + C(n,1) + C(n,2) + 3*C(n,4) - 3*C(n,5).
G.f.: -(-1+4*x-7*x^2+7*x^3-7*x^4+7*x^5)/(-1+x)^6. - R. J. Mathar, Jun 18 2009
a(0)=1, a(1)=2, a(2)=4, a(3)=7, a(4)=14, a(5)=28, a(n)=6*a(n-1)- 15*a(n-2)+ 20*a(n-3)-15*a(n-4)+6*a(n-5)-a(n-6). - Harvey P. Dale, Jan 14 2014

A133028 Even perfect numbers divided by 2.

Original entry on oeis.org

3, 14, 248, 4064, 16775168, 4294934528, 68719345664, 1152921504069976064, 1329227995784915872327346307976921088, 95780971304118053647396689042151819065498660774084608, 6582018229284824168619876730229361455111736159193471558891864064, 7237005577332262213973186563042994240786838745737417944533177174565599576064
Offset: 1

Views

Author

Omar E. Pol, Oct 20 2007, Apr 23 2008, Apr 28 2009

Keywords

Comments

a(13) has 314 digits and is too large to include. - R. J. Mathar, Oct 23 2007
Largest proper divisor of n-th even perfect number.
Also numbers k such that A000203(k) is divisible 24. - Ctibor O. Zizka, Jun 29 2009

Crossrefs

Programs

  • Maple
    a:=proc(n) if isprime(2^n-1)=true then 2^(n-2)*(2^n-1) else end if end proc: seq(a(n),n=1..120); # Emeric Deutsch, Oct 24 2007
  • Mathematica
    p = Select[2^Range[400] - 1, PrimeQ]; p*(p+1)/4 (* Vladimir Joseph Stephan Orlovsky, Feb 02 2012 *)
    Map[2^(#-2) * (2^# - 1) &, MersennePrimeExponent[Range[12]]] (* Amiram Eldar, Oct 21 2024 *)

Formula

a(n) = A000396(n)/2. - R. J. Mathar, Oct 23 2007 [Assuming there are no odd perfect numbers. - Jianing Song, Sep 17 2022]
a(n) = 2^(A000043(n) - 2) * A000668(n). - Omar E. Pol, Mar 01 2008
a(n) = A032742(A000396(n)), assuming there are no odd perfect numbers.

Extensions

More terms from R. J. Mathar and Emeric Deutsch, Oct 23 2007

A135652 Divisors of 28 (the 2nd perfect number), written in base 2.

Original entry on oeis.org

1, 10, 100, 111, 1110, 11100
Offset: 1

Views

Author

Omar E. Pol, Feb 23 2008, Mar 03 2008

Keywords

Comments

The number of divisors of the second perfect number is equal to 2*A000043(2)=A061645(2)=6.

Examples

			The structure of divisors of 28 (see A018254)
----------------------------------------------------------------------
n ... Divisor . Formula ....... Divisor written in base 2 ............
----------------------------------------------------------------------
1)......... 1 = 2^0 ........... 1
2)......... 2 = 2^1 ........... 10
3)......... 4 = 2^2 ........... 100 .... (The 2nd superperfect number)
4)......... 7 = 2^3 - 2^0 ..... 111 .... (The 2nd Mersenne prime)
5)........ 14 = 2^4 - 2^1 ..... 1110
6)........ 28 = 2^5 - 2^2 ..... 11100... (The 2nd perfect number)
		

Crossrefs

For more information see A018254 (Divisors of 28). Cf. A000043, A000079, A000396, A000668, A019279, A061645, A061652.

Programs

Formula

a(n)=A018254(n), written in base 2. Also, for n=1 .. 6: If n<=(A000043(2)=3) then a(n) is the concatenation of the digit "1" and n-1 digits "0" else a(n) is the concatenation of A000043(2)=3 digits "1" and (n-1-A000043(2)) digits "0".

A133031 Triangle read by rows: row n lists divisors of n-th perfect number A000396(n).

Original entry on oeis.org

1, 2, 3, 6, 1, 2, 4, 7, 14, 28, 1, 2, 4, 8, 16, 31, 62, 124, 248, 496, 1, 2, 4, 8, 16, 32, 64, 127, 254, 508, 1016, 2032, 4064, 8128, 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8191, 16382, 32764, 65528, 131056, 262112, 524224, 1048448, 2096896
Offset: 1

Views

Author

Omar E. Pol, Oct 26 2007

Keywords

Crossrefs

Cf. A018254, A018487, A027750. Perfect numbers: A000396.

Programs

  • Mathematica
    Divisors[PerfectNumber[Range[5]]]//Flatten (* Harvey P. Dale, Jul 29 2024 *)

A133024 Divisors of 8128, the 4th perfect number.

Original entry on oeis.org

1, 2, 4, 8, 16, 32, 64, 127, 254, 508, 1016, 2032, 4064, 8128
Offset: 1

Views

Author

Omar E. Pol, Oct 26 2007, Mar 03 2008, Dec 27 2008

Keywords

Comments

127 is the 4th Mersenne prime: A000668.
The number of divisors of the 4th perfect number is 2*A000043(4)=A061645(4)=14.
For the structure of this sequence and its binary expansion, see A135654.

Crossrefs

Cf. A018254, A018487. Perfect numbers: A000396.

Programs

Formula

For n=1..7 : a(n) = 2^(n-1). For n=8..14: a(n) = 2^(n-1) - 2^(n-8) = A000668(4)*2^(n-8).

A133025 Divisors of 33550336, the 5th perfect number.

Original entry on oeis.org

1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8191, 16382, 32764, 65528, 131056, 262112, 524224, 1048448, 2096896, 4193792, 8387584, 16775168, 33550336
Offset: 1

Views

Author

Omar E. Pol, Oct 26 2007, Mar 03 2008, Dec 27 2008

Keywords

Comments

8191 is the 5th Mersenne prime: A000668.
The number of divisors of the 5th perfect number is 2*A000043(5)=A061645(5)=26.
For the structure of this sequence and its binary expansion, see A135655.

Crossrefs

Cf. A018254, A018487. Perfect numbers: A000396.

Programs

Formula

For n=1..13 : a(n) = 2^(n-1). For n=14..26: a(n) = 2^(n-1) - 2^(n-14) = A000668(5)*2^(n-14).

A139247 Triangle read by rows: row n lists the divisors of n-th perfect number A000396(n) that are multiples of n-th Mersenne prime A000668(n).

Original entry on oeis.org

3, 6, 7, 14, 28, 31, 62, 124, 248, 496, 127, 254, 508, 1016, 2032, 4064, 8128, 8191, 16382, 32764, 65528, 131056, 262112, 524224, 1048448, 2096896, 4193792, 8387584, 16775168, 33550336, 131071, 262142, 524284, 1048568, 2097136, 4193792
Offset: 1

Views

Author

Omar E. Pol, Apr 22 2008

Keywords

Comments

Also, row n list the divisors of n-th perfect number that are not powers of 2.
First term of row n is the n-th Mersenne prime A000668(n). Last term of row n is the n-th perfect number A000396(n). Row n has A000043(n) terms. The sum of row n is equal to A133049(n), the square of n-th Mersenne prime A000668(n).

Examples

			Triangle begins:
  3, 6,
  7, 14, 28
  31, 62, 124, 248, 496
  127, 254, 508, 1016, 2032, 4064, 8128
  ...
==========================================================
Row .... First term ..... Last term ....... Row sum ......
n ..... (A000668(n)) ... (A000396(n)) ... (A000668(n)^2) .
==========================================================
1 ............ 3 .............. 6 ......... 3^2 = 9
2 ............ 7 ............. 28 ......... 7^2 = 49
3 ........... 31 ............ 496 ........ 31^2 = 961
4 .......... 127 ........... 8128 ....... 127^2 = 16129
5 ......... 8191 ....... 33550336 ...... 8191^2 = 67092481
		

Crossrefs

A241031 Sum of n-th powers of divisors of 28.

Original entry on oeis.org

6, 56, 1050, 25112, 655746, 17766056, 489541650, 13599182072, 379283617986, 10599157616456, 296486304875250, 8297561014164632, 232274972859656226, 6502905234227329256, 182070232515259616850, 5097810928082584052792
Offset: 0

Views

Author

Vincenzo Librandi, Apr 17 2014

Keywords

Crossrefs

Cf. A018254 (divisors of 28).
Cf. similar sequence listed in A241029.

Programs

  • Magma
    [DivisorSigma(n, 28): n in [0..20]];
    
  • Magma
    [(4^n+2^n+1)*(7^n+1): n in [0..20]];
  • Mathematica
    Total[#^Range[0, 20] & /@ Divisors[28]] (* or *) Table[(4^n + 2^n + 1) (7^n + 1), {n, 0, 20}]

Formula

G.f.: 2*(3 - 140*x + 2086*x^2 - 12360*x^3 + 29204*x^4 - 21952*x^5)/((1 - x)*(1 - 2*x)*(1 - 4*x)*(1 - 7*x)*(1 - 14*x)*(1 - 28*x)).
a(n) = (4^n + 2^n + 1)*(7^n + 1).

A139246 Triangle read by rows: row n lists the proper divisors of n-th perfect number A000396(n).

Original entry on oeis.org

1, 2, 3, 1, 2, 4, 7, 14, 1, 2, 4, 8, 16, 31, 62, 124, 248, 1, 2, 4, 8, 16, 32, 64, 127, 254, 508, 1016, 2032, 4064, 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8191, 16382, 32764, 65528, 131056, 262112, 524224, 1048448, 2096896, 4193792, 8387584, 16775168, 1
Offset: 1

Views

Author

Omar E. Pol, Apr 22 2008, corrected Apr 25 2008

Keywords

Comments

Rows n has A133033(n) terms.
The n-th row sum is the n-th perfect number A000396(n).

Examples

			Triangle begins:
  1, 2, 3
  1, 2, 4, 7, 14
  1, 2, 4, 8, 16, 31, 62, 124, 248
  1, 2, 4, 8, 16, 32, 64, 127, 254, 508, 1016, 2032, 4064
  ...
		

Crossrefs

Programs

  • Mathematica
    Table[Most[Divisors[PerfectNumber[n]]],{n,6}]//Flatten (* Harvey P. Dale, Jul 08 2024 *)

A139248 Triangle read by rows: row n lists the proper divisors of n-th even superperfect number A061652(n).

Original entry on oeis.org

1, 1, 2, 1, 2, 4, 8, 1, 2, 4, 8, 16, 32, 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384, 32768, 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384, 32768, 65536, 131072, 1, 2, 4, 8, 16
Offset: 1

Views

Author

Omar E. Pol, Apr 22 2008

Keywords

Comments

Also, row n list the proper divisors of n-th superperfect number A019279(n), if there are no odd superperfect numbers.
Row n has A000043(n) - 1 = A090748(n) terms.

Examples

			Triangle begins:
  1
  1, 2
  1, 2, 4, 8
  1, 2, 4, 8, 16, 32
  1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048
  1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384, 32768
  ...
		

Crossrefs

Showing 1-10 of 11 results. Next