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.

A171058 Numbers k such that 3^k + 3*k + 1 is prime.

Original entry on oeis.org

0, 1, 3, 41, 45, 83, 95, 181, 193, 195, 311, 325, 901, 1825, 3649, 7709, 16335, 45547
Offset: 1

Views

Author

N. J. A. Sloane, Sep 04 2010, based on a posting to the Sequence Fans Mailing List by Charles R Greathouse IV, Aug 23 2010

Keywords

Comments

a(19) > 10^5. - Michael S. Branicky, Jul 25 2024

Crossrefs

Numbers k such that b^k + b*k + 1 is prime: A301634 (b=2), this sequence (b=3), A301635 (b=5).
Cf. A176805, A180269.

Programs

Extensions

More terms from Charles R Greathouse IV, Sep 07 2010

A176916 a(n) = 5^n + 5*n + 1.

Original entry on oeis.org

2, 11, 36, 141, 646, 3151, 15656, 78161, 390666, 1953171, 9765676, 48828181, 244140686, 1220703191, 6103515696, 30517578201, 152587890706, 762939453211, 3814697265716, 19073486328221, 95367431640726, 476837158203231, 2384185791015736, 11920928955078241, 59604644775390746
Offset: 0

Views

Author

Jonathan Vos Post, Apr 28 2010

Keywords

Examples

			a(3) = 5^3 + 5*3 + 1 = 141.

Links

Crossrefs

Cf. A000351, A008587, A016861, A176691, A176805.

Programs

Formula

a(n) = A000351(n) + A008587(n) + 1 = A000351(n) + A016861(n).
From R. J. Mathar, Apr 29 2010: (Start)
a(n) = 7*a(n-1) - 11*a(n-2) + 5*a(n-3).
G.f.: ( -2+3*x+19*x^2 ) / ( (5*x-1)*(x-1)^2 ). (End)
E.g.f.: exp(x)*(1 + exp(4*x) + 5*x). - Stefano Spezia, Aug 19 2024

Extensions

First term corrected by several authors, Apr 29 2010
a(22)-a(24) from Stefano Spezia, Aug 19 2024

A176972 a(n) = 7^n + 7*n + 1.

Original entry on oeis.org

2, 15, 64, 365, 2430, 16843, 117692, 823593, 5764858, 40353671, 282475320, 1977326821, 13841287286, 96889010499, 678223072948, 4747561510049, 33232930569714, 232630513987327, 1628413597910576, 11398895185373277, 79792266297612142, 558545864083284155, 3909821048582988204
Offset: 0

Views

Author

Jonathan Vos Post, Apr 29 2010

Keywords

Examples

			a(5) = 7^5 + 7*5 + 1 = 16843 is prime.

Links

Crossrefs

Cf. A000420, A008589, A016993, A176691, A176805, A176916.
Cf. A370664.

Programs

  • Magma
    [7^n + 7*n + 1: n in [0..25]]; // Vincenzo Librandi, May 06 2011
  • Mathematica
    Table[7^n+7n+1,{n,0,20}] (* or *) LinearRecurrence[{9,-15,7},{2,15,64},20] (* Harvey P. Dale, Apr 17 2014 *)

Formula

a(n) = A000420(n) + A008589(n) + 1 = A000420(n) + A016993(n).
a(n) = 7*a(n-1) - 42*(n-1) + 1, with n > 0. For n=5, a(5) = 7*2430 - 42*4 + 1 = 16843. - Bruno Berselli, May 18 2010
From R. J. Mathar, May 22 2010: (Start)
a(n) = 9*a(n-1) - 15*a(n-2) + 7*a(n-3).
G.f.: (-2 + 3*x + 41*x^2) / ((7*x-1)*(x-1)^2). (End)
E.g.f.: exp(x)*(1 + exp(6*x) + 7*x). - Stefano Spezia, Aug 19 2024

A180269 Primes of the form 3^n+3*n+1.

Original entry on oeis.org

2, 7, 37, 36472996377170786527, 2954312706550833698779, 3990838394187339929534246675572349035477, 2120895147045314119491609587512844743630072393, 228532044137599177017869183161846685251274404207185590172004697234871412029099114059347
Offset: 1

Views

Author

Graziano Aglietti (mg5055(AT)mclink.it), Aug 23 2010

Keywords

Comments

Terms a(9) through a(12) have, respectively, 93, 94, 149, and 156 digits. - Harvey P. Dale, May 11 2013

Links

Crossrefs

Cf. A176805, A171058.

Programs

  • Mathematica
    Select[Table[3^n+3n+1,{n,0,100}],PrimeQ] (* Harvey P. Dale, May 11 2013 *)
  • PARI
    for(i=0,200,x=3^i+3*i+1;if(isprime(x),print1(x", ")))

Extensions

a(8) from Harvey P. Dale, May 11 2013

A221905 a(n) = 3^n + 3*n.

Original entry on oeis.org

1, 6, 15, 36, 93, 258, 747, 2208, 6585, 19710, 59079, 177180, 531477, 1594362, 4783011, 14348952, 43046769, 129140214, 387420543, 1162261524, 3486784461, 10460353266, 31381059675, 94143178896, 282429536553, 847288609518, 2541865828407, 7625597485068, 22876792455045
Offset: 0

Views

Author

Vincenzo Librandi, Mar 02 2013

Keywords

Links

Crossrefs

Cf. A107583, A176805, A370658.

Programs

  • Magma
    [3^n+3*n: n in [0..30]];
  • Magma
    I:=[1, 6, 15]; [n le 3 select I[n] else 5*Self(n-1)-7*Self(n-2)+3*Self(n-3): n in [1..30]];
  • Mathematica
    Table[(3^n + 3 n), {n, 0, 30}] (* or *) CoefficientList[Series[(1 + x - 8 x^2)/((1 - x)^2 (1 -3 x)), {x, 0, 30}], x]

Formula

G.f.: (1 + x - 8*x^2)/((1-x)^2*(1-3*x)).
a(n) = 5*a(n-1) - 7*a(n-2) + 3*a(n-3).
a(n) = A176805(n) - 1.
E.g.f.: exp(x)*(exp(2*x) + 3*x). - Elmo R. Oliveira, Sep 10 2024

A375577 Array read by ascending antidiagonals: A(n,k) = k^n + k*n + 1.

Original entry on oeis.org

2, 1, 2, 1, 3, 2, 1, 4, 5, 2, 1, 5, 9, 7, 2, 1, 6, 15, 16, 9, 2, 1, 7, 25, 37, 25, 11, 2, 1, 8, 43, 94, 77, 36, 13, 2, 1, 9, 77, 259, 273, 141, 49, 15, 2, 1, 10, 143, 748, 1045, 646, 235, 64, 17, 2, 1, 11, 273, 2209, 4121, 3151, 1321, 365, 81, 19, 2
Offset: 0

Views

Author

Stefano Spezia, Aug 19 2024

Keywords

Examples

			Array begins:
  2, 2,  2,   2,    2,     2, ...
  1, 3,  5,   7,    9,    11, ...
  1, 4,  9,  16,   25,    36, ...
  1, 5, 15,  37,   77,   141, ...
  1, 6, 25,  94,  273,   646, ...
  1, 7, 43, 259, 1045,  3151, ...
  1, 8, 77, 748, 4121, 15656, ...
  ...

Crossrefs

Cf. A000290, A004247, A004248, A005408 (n=1), A005491 (n=3), A007395 (n=0), A054977 (k=0), A176691 (k=2), A176805 (k=3), A176916 (k=5), A176972 (k=7), A214647.
Cf. A375578 (antidiagonal sums).

Programs

  • Mathematica
    A[0,0]=2; A[n_,k_]:=k^n+k*n+1;Table[A[n-k,k],{n,0,10},{k,0,n}]//Flatten

Formula

G.f. for the k-th column: (2*x^2 - 3*x - k^2 + k + 1)/((x - 1)^2*(x - k)).
E.g.f. for the k-th column: exp(x)*(1 + exp((k-1)*x) + k*x).
A(n,1) = n + 2.
A(2,n) = A000290(n+1).
A(n,n) = 2*A214647(n) + 1.
Showing 1-6 of 6 results.

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

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