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 1-4 of 4 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)

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)

A242341 Numbers k such that k*10^k - 1 is a semiprime.

Original entry on oeis.org

1, 6, 20, 29, 35, 40, 79, 164, 185, 198, 201, 218, 248, 249, 251, 264, 267, 274, 305, 323, 339, 344, 350, 362, 432, 539
Offset: 1

Views

Author

Vincenzo Librandi, May 12 2014

Keywords

Comments

The semiprimes of this form are: 9, 5999999, 1999999999999999999999, 2899999999999999999999999999999, ...
From Robert Israel, Sep 04 2016: (Start)
k == 1 (mod 3) is in the sequence iff (k*10^k-1)/3 is prime.
The sequence includes 185, 198, 201, 251, 267, 274, and 1795. (End)
a(27) >= 596. Below 1000, 785 and 833 are in the sequence. Unknown factorization for 596, 669, 917, 933. - Hugo Pfoertner, Jul 29 2019

Links

Crossrefs

Cf. similar sequences listed in A242273.
Cf. A059671, A064756.

Programs

  • Magma
    IsSemiprime:=func; [n: n in [2..70] | IsSemiprime(s) where s is n*10^n-1];
  • Maple
    issemiprime:= proc(n) local F, t;
    F:= ifactors(n, easy)[2];
    t:= add(f[2], f=F);
    if t = 1 then
       if type(F[1][1], integer) then return false fi
    elif t = 2 then
       return not hastype(F, name)
    else # t > 2
       return false
    fi;
    F:= ifactors(n)[2];
    return evalb(add(f[2], f=F)=2);
end proc:
select(t -> issemiprime(t*10^t-1), [$1..80]); # Robert Israel, Sep 04 2016
  • Mathematica
    Select[Range[70], PrimeOmega[# 10^# - 1]==2&]
  • PARI
    is(n)=bigomega(n*10^n-1)==2 \\ Charles R Greathouse IV, Sep 04 2016

Extensions

Terms 1 and 79 from Robert Israel, Sep 04 2016
a(8)-a(26) from Hugo Pfoertner, Jul 29 2019
Showing 1-4 of 4 results.

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

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