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.

A064753 a(n) = n*7^n - 1.

Original entry on oeis.org

6, 97, 1028, 9603, 84034, 705893, 5764800, 46118407, 363182462, 2824752489, 21750594172, 166095446411, 1259557135290, 9495123019885, 71213422649144, 531726889113615, 3954718737782518, 29311444762388081, 216579008522089716, 1595845325952240019, 11729463145748964146
Offset: 1

Views

Author

N. J. A. Sloane, Oct 19 2001

Keywords

Links

Crossrefs

For a(n)=n*k^n-1 cf. -A000012 (k=0), A001477 (k=1), A003261 (k=2), A060352 (k=3), A060416 (k=4), A064751 (k=5), A064752 (k=6), this sequence (k=7), A064754 (k=8), A064755 (k=9), A064756 (k=10), A064757 (k=11), A064758 (k=12).
Cf. A036293.

Programs

  • Magma
    [ n*7^n-1: n in [1..20]]; // Vincenzo Librandi, Sep 16 2011
  • Maple
    k:= 7; f:= gfun:-rectoproc({1 + (k-1)*n + k*n*a(n-1) - (n-1)*a(n) = 0, a(1) = k-1}, a(n), remember): map(f, [$1..20]); # Georg Fischer, Feb 19 2021
  • Mathematica
    Table[n 7^n-1,{n,20}] (* or *) LinearRecurrence[{15,-63,49},{6,97,1028},20] (* Harvey P. Dale, Feb 12 2022 *)

Formula

From Alois P. Heinz, Feb 19 2021: (Start)
G.f.: (56*x^2-21*x+1)/((x-1)*(7*x-1)^2).
a(n) = A036293(n) - 1. (End)
From Elmo R. Oliveira, May 05 2025: (Start)
E.g.f.: 1 + exp(x)*(7*x*exp(6*x) - 1).
a(n) = 15*a(n-1) - 63*a(n-2) + 49*a(n-3) for n > 3. (End)

A064756 a(n) = n*10^n - 1.

Original entry on oeis.org

9, 199, 2999, 39999, 499999, 5999999, 69999999, 799999999, 8999999999, 99999999999, 1099999999999, 11999999999999, 129999999999999, 1399999999999999, 14999999999999999, 159999999999999999, 1699999999999999999, 17999999999999999999, 189999999999999999999, 1999999999999999999999
Offset: 1

Views

Author

N. J. A. Sloane, Oct 19 2001

Keywords

Links

Crossrefs

Cf. for a(n) = n*k^n - 1: -A000012 (k=0), A001477 (k=1), A003261 (k=2), A060352 (k=3), A060416 (k=4), A064751 (k=5), A064752 (k=6), A064753 (k=7), A064754 (k=8), A064755 (k=9), this sequence (k=10), A064757 (k=11), A064758 (k=12).
Cf. A064748, A126431.

Programs

  • Magma
    [ n*10^n-1: n in [1..20]]; // Vincenzo Librandi, Sep 16 2011
  • Maple
    k:= 10; f:= gfun:-rectoproc({1 + (k-1)*n + k*n*a(n-1) - (n-1)*a(n) = 0, a(1) = k-1}, a(n), remember): map(f, [$1..20]); # Georg Fischer, Feb 19 2021
  • Mathematica
    Array[# 10^# - 1 &, 18] (* Michael De Vlieger, Jan 14 2020 *)

Formula

From Elmo R. Oliveira, Sep 07 2024: (Start)
G.f.: x*(100*x^2 - 10*x - 9)/((x - 1)*(10*x - 1)^2).
E.g.f.: 1 + exp(x)*(10*x*exp(9*x) - 1).
a(n) = 21*a(n-1) - 120*a(n-2) + 100*a(n-3) for n > 3.
a(n) = A126431(n) - 1 = A064748(n) - 2. (End)

A064757 a(n) = n*11^n - 1.

Original entry on oeis.org

10, 241, 3992, 58563, 805254, 10629365, 136410196, 1714871047, 21221529218, 259374246009, 3138428376720, 37661140520651, 448795257871102, 5316497670165373, 62658722541234764, 735195677817154575, 8592599484487994106, 100078511642860166657, 1162022718519876379528
Offset: 1

Views

Author

N. J. A. Sloane, Oct 19 2001

Keywords

Comments

Conjecture: satisfies a linear recurrence having signature (23,-143,121). - Harvey P. Dale, May 12 2019
This conjecture is true since a(n) - a(n-1) yields the recurrence 1 + 10*n + 11*n*a(n-1) - (n-1)*a(n) = 0 with polynomial coefficients in n. - Georg Fischer, Feb 19 2021

Links

Crossrefs

Cf. for a(n) = n*k^n - 1: -A000012(k=0), A001477(k=1), A003261 (k=2), A060352 (k=3), A060416 (k=4), A064751 (k=5), A064752 (k=6), A064753 (k=7), A064754 (k=8), A064755 (k=9), A064756 (k=10), this sequence (k=11), A064758 (k=12).
Cf. A064749.

Programs

  • Magma
    [n*11^n - 1: n in [1..20]]; // Vincenzo Librandi, Sep 16 2011
  • Maple
    k:= 11; f:= gfun:-rectoproc({1 + (k-1)*n + k*n*a(n-1) - (n-1)*a(n) = 0, a(1) = k-1}, a(n), remember): map(f, [$1..20]); # Georg Fischer, Feb 19 2021
  • Mathematica
    Table[n*11^n-1,{n,20}] (* Harvey P. Dale, May 12 2019 *)

Formula

From Elmo R. Oliveira, Sep 07 2024: (Start)
G.f.: x*(121*x^2 - 11*x - 10)/((x - 1)*(11*x - 1)^2).
E.g.f.: 1 + exp(x)*(11*x*exp(10*x) - 1).
a(n) = 23*a(n-1) - 143*a(n-2) + 121*a(n-3) for n > 3.
a(n) = A064749(n) - 2. (End)

A064758 a(n) = n*12^n - 1.

Original entry on oeis.org

11, 287, 5183, 82943, 1244159, 17915903, 250822655, 3439853567, 46438023167, 619173642239, 8173092077567, 106993205379071, 1390911669927935, 17974858503684095, 231105323618795519, 2958148142320582655, 37716388814587428863, 479219999055934390271, 6070119988041835610111, 76675199848949502443519
Offset: 1

Views

Author

N. J. A. Sloane, Oct 19 2001

Keywords

Links

Crossrefs

Cf. for a(n) = n*k^n - 1: -A000012(k=0), A001477(k=1), A003261 (k=2), A060352 (k=3), A060416 (k=4), A064751 (k=5), A064752 (k=6), A064753 (k=7), A064754 (k=8), A064755 (k=9), A064756 (k=10), A064757 (k=11), this sequence (k=12).
Cf. A064750.

Programs

  • Magma
    [n*12^n - 1: n in [1..30]]; // Vincenzo Librandi, Jun 21 2018
  • Mathematica
    CoefficientList[Series[(11 + 12 x - 144 x^2) / ((1 - 12 x)^2 (1 - x)), {x, 0, 33}], x] (* Vincenzo Librandi, Jun 21 2018 *)
  • PARI
    a(n) = { n*12^n - 1 } \\ Harry J. Smith, Sep 24 2009

Formula

G.f.: x*(11 + 12*x - 144*x^2)/((1 - 12*x)^2*(1 - x)). - Vincenzo Librandi, Jun 21 2018
From Elmo R. Oliveira, Sep 07 2024: (Start)
E.g.f.: 1 + exp(x)*(12*x*exp(11*x) - 1).
a(n) = 25*a(n-1) - 168*a(n-2) + 144*a(n-3) for n > 3.
a(n) = A064750(n) - 2. (End)

A242335 Numbers k such that k*4^k-1 is semiprime.

Original entry on oeis.org

10, 12, 18, 24, 27, 44, 47, 65, 71, 82, 84, 131, 134, 138, 143, 155, 164, 168, 197, 212, 227, 243, 248, 293, 302, 384, 401
Offset: 1

Views

Author

Vincenzo Librandi, May 12 2014

Keywords

Comments

The semiprimes of this form are: 10485759, 201326591, 1236950581247, 6755399441055743, 486388759756013567, 13617340432139183023890366463, ...

Crossrefs

Cf. similar sequences listed in A242273.
Cf. A060416, A086661.

Programs

  • Magma
    IsSemiprime:=func; [n: n in [2..100] | IsSemiprime(s) where s is n*4^n-1];
  • Mathematica
    Select[Range[100], PrimeOmega[# 4^# - 1]==2&]
  • PARI
    isok(n)=bigomega(n*4^n-1)==2 \\ Anders Hellström, Aug 18 2015

Extensions

a(12)-a(16) from Carl Schildkraut, Aug 18 2015
a(17)-a(27) from Charles R Greathouse IV, Aug 18 2015

A100689 a(n) = prime(n) * 4^prime(n) - 1.

Original entry on oeis.org

31, 191, 5119, 114687, 46137343, 872415231, 292057776127, 5222680231935, 1618481116086271, 8358680908399640575, 142962266571249025023, 698910239464707491627007, 198263834416799184651812863, 3326963855579459488791396351, 930930909542605966724141416447
Offset: 1

Views

Author

Parthasarathy Nambi, Dec 07 2004

Keywords

Examples

			a(1) = 2*4^2 - 1 = 31.

Programs

  • Magma
    [NthPrime(n)*4^NthPrime(n) - 1: n in [1..20]]; // Vincenzo Librandi, Aug 27 2015
  • Mathematica
    Do[Print[Prime[n]*4^(Prime[n]) - 1], {n, 1, 20}] (* Stefan Steinerberger, Feb 15 2006 *)
# 4^#-1&/@Prime[Range[20]] (* Harvey P. Dale, Dec 06 2012 *)
  • PARI
    vector(20, n, p=prime(n); p*4^p - 1) \\ Michel Marcus, Aug 27 2015

Formula

a(n) = A060416(prime(n)) = A060416(A000040(n)). - Michel Marcus, Aug 27 2015

Extensions

More terms from Stefan Steinerberger, Feb 15 2006
a(14)-a(15) from Vincenzo Librandi, Aug 27 2015
Showing 1-6 of 6 results.

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

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