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 1-5 of 5 results.

A226199 a(n) = 7^n + n.

Original entry on oeis.org

1, 8, 51, 346, 2405, 16812, 117655, 823550, 5764809, 40353616, 282475259, 1977326754, 13841287213, 96889010420, 678223072863, 4747561509958, 33232930569617, 232630513987224, 1628413597910467, 11398895185373162, 79792266297612021, 558545864083284028, 3909821048582988071
Offset: 0

Views

Author

Vincenzo Librandi, Jun 16 2013

Keywords

Comments

Smallest prime of this form is a(34) = 54116956037952111668959660883.
In general, the g.f. of a sequence of numbers of the form k^n + n is (1-x-(k-1)*x^2)/((1-k*x)*(x-1)^2) with main linear recurrence (k+2)*a(n-1) - (2*k+1)*a(n-2) + k*a(n-3). - Bruno Berselli, Jun 16 2013

Links

Crossrefs

Cf. numbers of the form k^n + n: A006127 (k=2), A104743 (k=3), A158879 (k=4), A104745 (k=5), A226200 (k=6), this sequence (k=7), A226201 (k=8), A226202 (k=9), A081552 (k=10), A226737 (k=11).
Cf. A199483 (first differences), A370657.

Programs

  • Magma
    [7^n+n: n in [0..20]];
  • Magma
    I:=[1, 8, 51]; [n le 3 select I[n] else 9*Self(n-1)-15*Self(n-2)+7*Self(n-3): n in [1..30]];
  • Mathematica
    Table[7^n + n, {n, 0, 30}] (* or *) CoefficientList[Series[(1 - x - 6 x^2) / ((1 - 7 x) (1 - x)^2), {x, 0, 20}], x]
LinearRecurrence[{9,-15,7},{1,8,51},30] (* Harvey P. Dale, Jun 16 2025 *)
  • PARI
    a(n)=7^n+n \\ Charles R Greathouse IV, Oct 07 2015

Formula

G.f.: (1-x-6*x^2)/((1-7*x)*(1-x)^2).
a(n) = 9*a(n-1) - 15*a(n-2) + 7*a(n-3).
E.g.f.: exp(x)*(exp(6*x) + x). - Elmo R. Oliveira, Mar 05 2025

A226201 a(n) = 8^n + n.

Original entry on oeis.org

1, 9, 66, 515, 4100, 32773, 262150, 2097159, 16777224, 134217737, 1073741834, 8589934603, 68719476748, 549755813901, 4398046511118, 35184372088847, 281474976710672, 2251799813685265, 18014398509482002, 144115188075855891, 1152921504606846996, 9223372036854775829
Offset: 0

Views

Author

Vincenzo Librandi, Jun 16 2013

Keywords

Comments

Smallest prime of this form is a(101). - Bruno Berselli, Jun 17 2013

Links

Crossrefs

Cf. numbers of the form k^n+n: A006127 (k=2), A104743 (k=3), A158879 (k=4), A104745 (k=5), A226200 (k=6), A226199 (k=7), this sequence (k=8), A226202 (k=9), A081552 (k=10), A226737 (k=11).
Cf. A199555 (first differences).

Programs

  • Magma
    [8^n+n: n in [0..30]];
  • Magma
    I:=[1, 9, 66]; [n le 3 select I[n] else 10*Self(n-1)-17*Self(n-2)+8*Self(n-3): n in [1..30]];
  • Mathematica
    Table[8^n + n, {n, 0, 30}] (* or *) CoefficientList[Series[(-1 + x + 7 x^2) / ((8 x - 1) (x - 1)^2), {x, 0, 30}], x]
LinearRecurrence[{10,-17,8},{1,9,66},30] (* Harvey P. Dale, Aug 11 2015 *)
  • PARI
    a(n)=8^n+n \\ Charles R Greathouse IV, Oct 07 2015

Formula

G.f.: (-1+x+7*x^2)/((8*x-1)*(x-1)^2).
a(n) = 10*a(n-1) - 17*a(n-2) + 8*a(n-3).
E.g.f.: exp(x)*(exp(7*x) + x). - Elmo R. Oliveira, Mar 05 2025

A226202 a(n) = 9^n + n.

Original entry on oeis.org

1, 10, 83, 732, 6565, 59054, 531447, 4782976, 43046729, 387420498, 3486784411, 31381059620, 282429536493, 2541865828342, 22876792454975, 205891132094664, 1853020188851857, 16677181699666586, 150094635296999139, 1350851717672992108, 12157665459056928821, 109418989131512359230
Offset: 0

Views

Author

Vincenzo Librandi, Jun 16 2013

Keywords

Comments

After 83, the next prime of this form is a(76). - Bruno Berselli, Jun 18 2013

Links

Crossrefs

Cf. numbers of the form k^n + n: A006127 (k=2), A104743 (k=3), A158879 (k=4), A104745 (k=5), A226200 (k=6), A226199 (k=7), A226201 (k=8), this sequence (k=9), A081552 (k=10), A226737 (k=11).
Cf. A199677 (first differences).

Programs

  • Magma
    [9^n+n: n in [0..30]];
  • Magma
    I:=[1, 10, 83]; [n le 3 select I[n] else 11*Self(n-1)-19*Self(n-2)+9*Self(n-3): n in [1..30]];
  • Mathematica
    Table[9^n + n, {n, 0, 30}] (* or *) CoefficientList[Series[(- 1 + x + 8 x^2) / ((9 x - 1) (x - 1)^2), {x, 0, 30}], x]
LinearRecurrence[{11,-19,9},{1,10,83},20] (* Harvey P. Dale, Feb 03 2016 *)
  • PARI
    a(n)=9^n+n \\ Charles R Greathouse IV, Oct 07 2015

Formula

G.f.: (-1+x+8*x^2)/((9*x-1)*(x-1)^2).
a(n) = 11*a(n-1) - 19*a(n-2) + 9*a(n-3).
E.g.f.: exp(x)*(exp(8*x) + x). - Elmo R. Oliveira, Sep 09 2024

A226737 a(n) = 11^n + n.

Original entry on oeis.org

1, 12, 123, 1334, 14645, 161056, 1771567, 19487178, 214358889, 2357947700, 25937424611, 285311670622, 3138428376733, 34522712143944, 379749833583255, 4177248169415666, 45949729863572177, 505447028499293788, 5559917313492231499, 61159090448414546310, 672749994932560009221
Offset: 0

Views

Author

Vincenzo Librandi, Jun 16 2013

Keywords

Links

Crossrefs

Cf. numbers of the form k^n + n: A006127 (k=2), A104743 (k=3), A158879 (k=4), A104745 (k=5), A226200 (k=6), A226199 (k=7), A226201 (k=8), A226202 (k=9), A081552 (k=10), this sequence (k=11).
Cf. A199764 (first differences).

Programs

  • Magma
    [11^n+n: n in [0..30]];
  • Magma
    I:=[1, 12, 123]; [n le 3 select I[n] else 13*Self(n-1)-23*Self(n-2)+11*Self(n-3): n in [1..30]];
  • Mathematica
    Table[11^n + n, {n, 0, 30}] (* or *) CoefficientList[Series[(- 1 + x + 10 x^2) / ((11 x - 1) (x - 1)^2), {x, 0, 30}], x]
LinearRecurrence[{13,-23,11},{1,12,123},20] (* Harvey P. Dale, Nov 14 2018 *)
  • PARI
    a(n)=11^n+n \\ Charles R Greathouse IV, Oct 07 2015

Formula

G.f.: (-1+x+10*x^2)/((11*x-1)*(x-1)^2).
a(n) = 13*a(n-1) - 23*a(n-2) + 11*a(n-3).
E.g.f.: exp(x)*(exp(10*x) + x). - Elmo R. Oliveira, Mar 06 2025

A370190 Numbers k such that (6^k + 6*k)/6 is prime.

Original entry on oeis.org

1, 5, 7, 1003, 40513
Offset: 1

Views

Author

Hugo Pfoertner, Feb 24 2024

Keywords

Comments

If it exists, a(6) > 62000.

Crossrefs

Cf. A058828, A221908, A226200.

Programs

  • PARI
    is(n) = ispseudoprime(6^(n-1)+n)
Showing 1-5 of 5 results.

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