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 10 results.

A024037 a(n) = 4^n - n.

Original entry on oeis.org

1, 3, 14, 61, 252, 1019, 4090, 16377, 65528, 262135, 1048566, 4194293, 16777204, 67108851, 268435442, 1073741809, 4294967280, 17179869167, 68719476718, 274877906925, 1099511627756, 4398046511083, 17592186044394, 70368744177641, 281474976710632, 1125899906842599
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Cf. numbers of the form k^n - n: A000325 (k=2), A024024 (k=3), this sequence (k=4), A024050 (k=5), A024063 (k=6), A024076 (k=7), A024089 (k=8), A024102 (k=9), A024115 (k=10), A024128 (k=11), A024141 (k=12).
Cf. A140660 (first differences).

Programs

  • Magma
    [4^n - n: n in [0..35]]; // Vincenzo Librandi, May 13 2011
  • Magma
    I:=[1, 3, 14]; [n le 3 select I[n] else 6*Self(n-1)-9*Self(n-2)+4*Self(n-3): n in [1..30]]; // Vincenzo Librandi, Jun 16 2013
  • Mathematica
    Table[4^n - n, {n, 0, 30}] (* or *) CoefficientList[Series[(1 - 3 x + 5 x^2) / ((1 - 4 x) (1 - x)^2), {x, 0, 30}], x] (* Vincenzo Librandi, Jun 16 2013 *)
  • PARI
    a(n)=4^n-n \\ Charles R Greathouse IV, Sep 24 2015

Formula

From Vincenzo Librandi, Jun 16 2013: (Start)
G.f.: (1 - 3*x + 5*x^2)/((1 - 4*x)*(1 - x)^2).
a(n) = 6*a(n-1) - 9*a(n-2) + 4*a(n-3). (End)
E.g.f.: exp(x)*(exp(3*x) - x). - Elmo R. Oliveira, Sep 10 2024

A024050 a(n) = 5^n - n.

Original entry on oeis.org

1, 4, 23, 122, 621, 3120, 15619, 78118, 390617, 1953116, 9765615, 48828114, 244140613, 1220703112, 6103515611, 30517578110, 152587890609, 762939453108, 3814697265607, 19073486328106, 95367431640605, 476837158203104, 2384185791015603, 11920928955078102, 59604644775390601
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Cf. numbers of the form k^n - n: A000325 (k=2), A024024 (k=3), A024037 (k=4), this sequence (k=5), A024063 (k=6), A024076 (k=7), A024089 (k=8), A024102 (k=9), A024115 (k=10), A024128 (k=11), A024141 (k=12).

Programs

  • Magma
    [5^n-n: n in [0..35]]; // Vincenzo Librandi, Jun 12 2011
  • Maple
    g:=1/(1-5*z): gser:=series(g, z=0, 43): seq(coeff(gser, z, n)-n, n=0..31); # Zerinvary Lajos, Jan 09 2009
  • Mathematica
    Table[5^n - n, {n, 0, 30}] (* or *) CoefficientList[Series[(1 - 3 x + 6 x^2) / ((1 - 5 x) (1 - x)^2), {x, 0, 30}], x] (* Vincenzo Librandi, Jun 16 2013 *)
LinearRecurrence[{7,-11,5},{1,4,23},30] (* Harvey P. Dale, Mar 03 2022 *)

Formula

a(n) = 7*a(n-1) - 11*a(n-2) + 5*a(n-3).
G.f.: (1 - 3*x + 6*x^2)/((1-5*x)*(1-x)^2). - Vincenzo Librandi, Jun 16 2013
E.g.f.: exp(x)*(exp(4*x) - x). - Elmo R. Oliveira, Sep 10 2024

A024063 a(n) = 6^n - n.

Original entry on oeis.org

1, 5, 34, 213, 1292, 7771, 46650, 279929, 1679608, 10077687, 60466166, 362797045, 2176782324, 13060694003, 78364164082, 470184984561, 2821109907440, 16926659444719, 101559956668398, 609359740010477, 3656158440062956, 21936950640377835, 131621703842267114, 789730223053602793
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Cf. numbers of the form k^n - n: A000325 (k=2), A024024 (k=3), A024037 (k=4), A024050 (k=5), this sequence (k=6), A024076 (k=7), A024089 (k=8), A024102 (k=9), A024115 (k=10), A024128 (k=11), A024141 (k=12).

Programs

  • Magma
    [6^n-n: n in [0..25]]; // Vincenzo Librandi, Jul 03 2011
  • Magma
    I:=[1, 5, 34]; [n le 3 select I[n] else 8*Self(n-1)-13*Self(n-2)+6*Self(n-3): n in [1..30]]; // Vincenzo Librandi, Jun 16 2013
  • Mathematica
    Table[6^n - n, {n, 0, 30}] (* or *) CoefficientList[Series[(1 - 3 x + 7 x^2) / ((1 - 6 x) (1 - x)^2), {x, 0, 30}], x] (* Vincenzo Librandi, Jun 16 2013 *)
  • PARI
    a(n)=6^n-n \\ Charles R Greathouse IV, Oct 07 2015

Formula

From Vincenzo Librandi, Jun 16 2013: (Start)
G.f.: (1-3*x+7*x^2)/((1-6*x)*(1-x)^2).
a(n) = 8*a(n-1) - 13*a(n-2) + 6*a(n-3). (End)
E.g.f.: exp(x)*(exp(5*x) - x). - Elmo R. Oliveira, Sep 10 2024

A024076 a(n) = 7^n - n.

Original entry on oeis.org

1, 6, 47, 340, 2397, 16802, 117643, 823536, 5764793, 40353598, 282475239, 1977326732, 13841287189, 96889010394, 678223072835, 4747561509928, 33232930569585, 232630513987190, 1628413597910431, 11398895185373124, 79792266297611981, 558545864083283986, 3909821048582988027
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Cf. numbers of the form k^n - n: A000325 (k=2), A024024 (k=3), A024037 (k=4), A024050 (k=5), A024063 (k=6), this sequence (k=7), A024089 (k=8), A024102 (k=9), A024115 (k=10), A024128 (k=11), A024141 (k=12).
Cf. A198688 (first differences).

Programs

Formula

From Vincenzo Librandi, Jun 16 2013: (Start)
G.f.: (1-3*x+8*x^2)/((1-7*x)*(1-x)^2).
a(n) = 9*a(n-1) - 15*a(n-2) + 7*a(n-3). (End)
E.g.f.: exp(x)*(exp(6*x) - x). - Elmo R. Oliveira, Sep 10 2024

A024089 a(n) = 8^n - n.

Original entry on oeis.org

1, 7, 62, 509, 4092, 32763, 262138, 2097145, 16777208, 134217719, 1073741814, 8589934581, 68719476724, 549755813875, 4398046511090, 35184372088817, 281474976710640, 2251799813685231, 18014398509481966, 144115188075855853, 1152921504606846956, 9223372036854775787
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Cf. numbers of the form k^n - n: A000325 (k=2), A024024 (k=3), A024037 (k=4), A024050 (k=5), A024063 (k=6), A024076 (k=7), this sequence (k=8), A024102 (k=9), A024115 (k=10), A024128 (k=11), A024141 (k=12).
Cf. A198855 (first differences).

Programs

  • Magma
    [8^n-n: n in [0..20]]; // Vincenzo Librandi, Jul 05 2011
  • Magma
    I:=[1,7,62]; [n le 3 select I[n] else 10*Self(n-1)-17*Self(n-2)+8*Self(n-3): n in [1..30]]; // Vincenzo Librandi, Jun 17 2013
  • Mathematica
    Table[8^n - n, {n, 0, 20}] (* or *) CoefficientList[Series[(1 - 3 x + 9 x^2) / ((1 - 8 x) (1 - x)^2), {x, 0, 30}], x] (* Vincenzo Librandi, Jun 17 2013 *)
LinearRecurrence[{10,-17,8},{1,7,62},30] (* Harvey P. Dale, Sep 28 2017 *)
  • PARI
    a(n)=8^n-n \\ Charles R Greathouse IV, Oct 07 2015

Formula

From Vincenzo Librandi, Jun 17 2013: (Start)
G.f.: (1-3*x+9*x^2)/((1-8*x)*(1-x)^2).
a(n) = 10*a(n-1) - 17*a(n-2) + 8*a(n-3). (End)
E.g.f.: exp(x)*(exp(7*x) - x). - Elmo R. Oliveira, Sep 10 2024

A024102 a(n) = 9^n - n.

Original entry on oeis.org

1, 8, 79, 726, 6557, 59044, 531435, 4782962, 43046713, 387420480, 3486784391, 31381059598, 282429536469, 2541865828316, 22876792454947, 205891132094634, 1853020188851825, 16677181699666552, 150094635296999103, 1350851717672992070, 12157665459056928781, 109418989131512359188
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Cf. numbers of the form k^n - n: A000325 (k=2), A024024 (k=3), A024037 (k=4), A024050 (k=5), A024063 (k=6), A024076 (k=7), A024089 (k=8), this sequence (k=9), A024115 (k=10), A024128 (k=11), A024141 (k=12).
Cf. A198966 (first differences).

Programs

  • Magma
    [9^n-n: n in [0..25]]; // Vincenzo Librandi, Jul 06 2011
  • Magma
    I:=[1, 8, 79]; [n le 3 select I[n] else 11*Self(n-1)-19*Self(n-2)+9*Self(n-3): n in [1..30]]; // Vincenzo Librandi, Jun 17 2013
  • Mathematica
    Table[9^n - n, {n, 0, 20}] (* or *) CoefficientList[Series[(1 - 3 x + 10 x^2) / ((1 - 9 x) (1 - x)^2), {x, 0, 30}], x] (* Vincenzo Librandi, Jun 17 2013 *)
LinearRecurrence[{11,-19,9},{1,8,79},30] (* Harvey P. Dale, Dec 25 2024 *)
  • PARI
    a(n)=9^n-n \\ Charles R Greathouse IV, Oct 07 2015

Formula

From Vincenzo Librandi, Jun 17 2013: (Start)
G.f.: (1-3*x+10*x^2)/((1-9*x)(1-x)^2).
a(n) = 11*a(n-1) - 19*a(n-2) + 9*a(n-3). (End)
E.g.f.: exp(x)*(exp(8*x) - x). - Elmo R. Oliveira, Sep 09 2024

A024128 a(n) = 11^n - n.

Original entry on oeis.org

1, 10, 119, 1328, 14637, 161046, 1771555, 19487164, 214358873, 2357947682, 25937424591, 285311670600, 3138428376709, 34522712143918, 379749833583227, 4177248169415636, 45949729863572145, 505447028499293754, 5559917313492231463, 61159090448414546272, 672749994932560009181
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

Smallest prime of this form is a(18) = 5559917313492231463. - Bruno Berselli, Jun 17 2013

Links

Crossrefs

Cf. numbers of the form k^n - n: A000325 (k=2), A024024 (k=3), A024037 (k=4), A024050 (k=5), A024063 (k=6), A024076 (k=7), A024089 (k=8), A024102 (k=9), A024115 (k=10), this sequence (k=11), A024141 (k=12).
Cf. A199030 (first differences).

Programs

  • Magma
    [11^n-n: n in [0..20]]; // Vincenzo Librandi, Jul 01 2011
  • Magma
    I:=[1, 10, 119]; [n le 3 select I[n] else 13*Self(n-1)-23*Self(n-2)+11*Self(n-3): n in [1..30]]; // Vincenzo Librandi, Jun 17 2013
  • Mathematica
    Table[11^n - n, {n, 0, 20}] (* or *) CoefficientList[Series[(1 - 3 x + 12 x^2) / ((1 - 11 x) (1 - x)^2), {x, 0, 30}], x] (* Vincenzo Librandi, Jun 17 2013 *)
LinearRecurrence[{13,-23,11},{1,10,119},20] (* Harvey P. Dale, Aug 02 2017 *)
  • PARI
    a(n)=11^n-n \\ Charles R Greathouse IV, Jul 01 2011

Formula

From Vincenzo Librandi, Jun 17 2013: (Start)
G.f.: (1-3*x+12*x^2)/((1-11*x) (1-x)^2).
a(n) = 13*a(n-1) - 23*a(n-2) + 11*a(n-3). (End)
E.g.f.: exp(x)*(exp(10*x) - x). - Elmo R. Oliveira, Sep 10 2024

A024141 a(n) = 12^n - n.

Original entry on oeis.org

1, 11, 142, 1725, 20732, 248827, 2985978, 35831801, 429981688, 5159780343, 61917364214, 743008370677, 8916100448244, 106993205379059, 1283918464548850, 15407021574586353, 184884258895036400, 2218611106740436975, 26623333280885243886, 319479999370622926829
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Cf. numbers of the form k^n - n: A000325 (k=2), A024024 (k=3), A024037 (k=4), A024050 (k=5), A024063 (k=6), A024076 (k=7), A024089 (k=8), A024102 (k=9), A024115 (k=10), A024128 (k=11), this sequence (k=12).

Programs

  • Magma
    [12^n-n: n in [0..20]]; // Vincenzo Librandi, Jul 01 2011
  • Magma
    I:=[1, 11, 142]; [n le 3 select I[n] else 14*Self(n-1)-25*Self(n-2)+12*Self(n-3): n in [1..30]]; // Vincenzo Librandi, Jun 17 2013
  • Mathematica
    Table[12^n - n, {n, 0, 20}] (* or *) CoefficientList[Series[(1 - 3 x + 13 x^2) / ((1 - 12 x) (1 - x)^2),{x, 0, 30}], x] (* Vincenzo Librandi, Jun 17 2013 *)
LinearRecurrence[{14,-25,12},{1,11,142},20] (* Harvey P. Dale, Nov 03 2024 *)
  • PARI
    a(n)=12^n-n \\ Charles R Greathouse IV, Jul 01 2011

Formula

From Vincenzo Librandi, Jun 17 2013: (Start)
G.f.: (1 - 3*x + 13*x^2)/((1-12*x)*(1-x)^2).
a(n) = 14*a(n-1) - 25*a(n-2) + 12*a(n-3). (End)
E.g.f.: exp(x)*(exp(11*x) - x). - Elmo R. Oliveira, Sep 10 2024

A110065 Numbers k such that 10^k - k is prime.

Original entry on oeis.org

3, 23, 171, 903, 9911, 48107, 48449, 60959
Offset: 1

Views

Author

Farideh Firoozbakht, Jul 11 2005

Keywords

Comments

For each n, a(n) == 3 (mod 6) or a(n) == 5 (mod 6).
There is no further term up to 16000. - Farideh Firoozbakht, Dec 02 2006
a(9) > 10^5. - Robert Price, Feb 19 2015

Crossrefs

Cf. A024115 (10^n-n).
Cf. A110066, A089379, A110067, A110068, A174175, A174176, A174177, A174178, A193881, A193882.

Programs

  • PARI
    is(n)=isprime(10^n-n) \\ Charles R Greathouse IV, Feb 17 2017
  • Python
    from sympy import isprime
def afind(limit):
  m, pow10 = 0, 1
  while m <= limit:
    if isprime(pow10 - m): print(m, end=", ")
    m, pow10 =  m + 1, pow10 * 10
afind(1000) # Michael S. Branicky, Mar 23 2021

Extensions

a(5) from Farideh Firoozbakht, Dec 02 2006
Definition corrected by Farideh Firoozbakht, Dec 12 2006
a(6)-a(8) from Robert Price, Feb 19 2015

A240091 a(n) = (10^n - 1)^n.

Original entry on oeis.org

1, 9, 9801, 997002999, 9996000599960001, 9999500009999900000499999, 999994000014999980000014999994000001, 9999993000002099999650000034999997900000069999999, 9999999200000027999999440000006999999944000000279999999200000001
Offset: 0

Views

Author

Hannah Ko, Mar 31 2014

Keywords

Examples

			a(3) = 999^3 = 997002999.

Crossrefs

Cf. A024115.

Extensions

Extended by N. J. A. Sloane, Apr 01 2014
Showing 1-10 of 10 results.

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

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