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 1-7 of 7 results.

A370657 Numbers k such that A226199(k) = 7^k + k is prime.

Original entry on oeis.org

34, 48, 8578, 18960
Offset: 1

Views

Author

Hugo Pfoertner, Feb 24 2024

Keywords

Comments

If it exists, a(5) > 268000. - Karl-Heinz Hofmann, Mar 02 2024

Crossrefs

Cf. A058828, A221909, A226199, A370660.

Programs

  • PARI
    is(n) = ispseudoprime(7^n+n)

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

A226200 a(n) = 6^n + n.

Original entry on oeis.org

1, 7, 38, 219, 1300, 7781, 46662, 279943, 1679624, 10077705, 60466186, 362797067, 2176782348, 13060694029, 78364164110, 470184984591, 2821109907472, 16926659444753, 101559956668434, 609359740010515, 3656158440062996, 21936950640377877, 131621703842267158, 789730223053602839
Offset: 0

Views

Author

Vincenzo Librandi, Jun 16 2013

Keywords

Comments

After 7, the next prime of this form has 238 digits (see A058828). - 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), this sequence (k=6), A226199 (k=7), A226201 (k=8), A226202 (k=9), A081552 (k=10), A226737 (k=11).
Cf. A058828, A199320 (first differences).

Programs

  • Magma
    [6^n+n: n in [0..30]];
  • Magma
    I:=[1, 7, 38]; [n le 3 select I[n] else 8*Self(n-1)-13*Self(n-2)+6*Self(n-3): n in [1..30]];
  • Mathematica
    Table[6^n + n, {n, 0, 30}] (* or *) CoefficientList[Series[(-1 + x + 5 x^2) / ((6 x - 1) (x - 1)^2), {x, 0, 30}], x]
  • PARI
    a(n)=6^n+n \\ Charles R Greathouse IV, Oct 07 2015

Formula

G.f.: (-1+x+5*x^2)/((6*x-1)*(x-1)^2).
a(n) = 8*a(n-1) - 13*a(n-2) + 6*a(n-3).
E.g.f.: exp(x)*(exp(5*x) + x). - Elmo R. Oliveira, Mar 05 2025

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

A370660 Numbers k such that (7^k + 7*k)/7 is prime.

Original entry on oeis.org

1, 4, 36, 160, 342, 1086, 2094, 11524
Offset: 1

Views

Author

Hugo Pfoertner, Feb 24 2024

Keywords

Comments

If it exists, a(11) > 50000.

Crossrefs

Cf. A221909, A226199, A370657.
Cf. A370190, A370661.

Programs

  • PARI
    is(n) = ispseudoprime(7^(n-1)+n)

A337455 Numbers of the form m + bigomega(m) with m a positive integer.

Original entry on oeis.org

1, 3, 4, 6, 8, 11, 12, 14, 15, 16, 17, 18, 20, 21, 23, 24, 27, 28, 30, 31, 32, 33, 35, 36, 37, 38, 40, 41, 42, 44, 45, 47, 48, 51, 53, 54, 55, 57, 58, 59, 60, 62, 64, 66, 67, 68, 69, 70, 71, 72, 73, 74, 76, 77, 78, 79, 80, 81, 84, 85, 87, 88, 89, 90, 92, 93
Offset: 1

Views

Author

Nathan J. McDougall, Aug 27 2020

Keywords

Comments

If a(n) = m + A001222(m) then (a(n) - m) <= log(a(n))/log(2).
It appears that a(n)/n may converge to a constant around ~ 1.49.

Examples

			a(7) = 10 + A001222(10) = 10 + 2 = 12

Links

Crossrefs

Cf. A001222 (bigomega), A064800, A358973.
Numbers of the form k^n+n where k is prime are subsequences: A006127 (k=2), A104743 (k=3), A104745 (k=5), A226199 (k=7), A226737 (k=11).
Subsequences include A008864, A101340, and A160649 (excluding the first term).

Programs

  • Mathematica
    m = 100; Select[Union @ Table[n + PrimeOmega[n], {n, 1, m}], # <= m &] (* Amiram Eldar, Aug 28 2020 *)
  • PARI
    upto(limit)=Set(select(t->t<=limit, apply(m->m+bigomega(m), [1..limit]))) \\ Andrew Howroyd, Aug 27 2020
  • PARI
    list(lim)=my(v=List()); forfactored(n=1, lim\1-1, my(t=n[1]+bigomega(n)); if(t<=lim, listput(v, t))); Set(v) \\ Charles R Greathouse IV, Dec 07 2022

Formula

Kucheriaviy proves that a(n) << n log log n and conjectures that a(n) ≍ n, that is, these numbers have positive lower density. - Charles R Greathouse IV, Dec 07 2022
Showing 1-7 of 7 results.

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