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.

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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

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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

A382786 Numbers k such that 5^k + k is prime.

Original entry on oeis.org

7954, 22102, 33054, 135156
Offset: 1

Views

Author

Nico Puada, Apr 24 2025

Keywords

Comments

All terms must end in the digit 2, 4, 6 or 8, otherwise 5^k + k is divisible by 2 or 5, which is not prime. - Jakub Buczak, May 04 2025

Crossrefs

Cf. A104745, A052007, A057900, A057909, A058046.

Programs

  • Mathematica
    Select[Range[35000], PrimeQ[5^# + #] &] (* Nico Puada, Apr 24 2025 *)

Extensions

a(4) corrected by Jakub Buczak, May 04 2025

A268414 a(n) = 5*a(n-1) - 2*n for n > 0, a(0) = 1.

Original entry on oeis.org

1, 3, 11, 49, 237, 1175, 5863, 29301, 146489, 732427, 3662115, 18310553, 91552741, 457763679, 2288818367, 11444091805, 57220458993, 286102294931, 1430511474619, 7152557373057, 35762786865245, 178813934326183, 894069671630871, 4470348358154309, 22351741790771497, 111758708953857435
Offset: 0

Views

Author

Ilya Gutkovskiy, Feb 04 2016

Keywords

Comments

In general, the ordinary generating function for the recurrence relation b(n) = k*b(n - 1) - m*n, with n > 0 and b(0)=1, is (1 - (m + 2)*x + x^2)/((1 - x)^2*(1 - k*x)). This recurrence gives the closed form b(n) = ((k^2 - k*(m + 2) + 1)*k^n + m*((k - 1)*n + k))/(k - 1)^2.

Links

Crossrefs

Cf. A014827, A024050, A094195, A104745, A107585, A164045, A176916, A221907.

Programs

  • Magma
    [(4*n + 3*5^n + 5)/8: n in [0..30]]; // Vincenzo Librandi, Feb 06 2016
  • Mathematica
    Table[(4 n + 3 5^n + 5)/8, {n, 0, 23}]
LinearRecurrence[{7, -11, 5}, {1, 3, 11}, 24]
  • PARI
    Vec((1-4*x+x^2)/((1-x)^2*(1-5*x)) + O(x^100)) \\ Altug Alkan, Feb 04 2016

Formula

G.f.: (1 - 4*x + x^2)/((1 - x)^2*(1 - 5*x)).
a(n) = (4*n + 3*5^n + 5)/8.
Sum_{n>=0} 1/a(n) = 1.449934283402232875...
Lim_{n -> oo} a(n + 1)/a(n) = 5.
From Elmo R. Oliveira, Sep 10 2024: (Start)
E.g.f.: exp(x)*(3*exp(4*x) + 4*x + 5)/8.
a(n) = 7*a(n-1) - 11*a(n-2) + 5*a(n-3) for n > 2. (End)

Extensions

a(24)-a(25) from Elmo R. Oliveira, Sep 10 2024
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