A104081 -id:A104081 - cp's OEIS frontend

A104080 Smallest prime >= 2^n.

2, 2, 5, 11, 17, 37, 67, 131, 257, 521, 1031, 2053, 4099, 8209, 16411, 32771, 65537, 131101, 262147, 524309, 1048583, 2097169, 4194319, 8388617, 16777259, 33554467, 67108879, 134217757, 268435459, 536870923, 1073741827, 2147483659

Offset: 0

Cino Hilliard, Mar 03 2005

Jinyuan Wang, Table of n, a(n) for n = 0..1000

Cf. A104081, A338475.
Except initial terms and offset, same as A014210 and A203074.
The opposite (greatest prime <= 2^n) is A014234, indices A007053.
The distance from 2^n is A092131, opposite A013603.
Counting zeros instead of both bits gives A372474, cf. A035103, A211997.
Counting ones instead of both bits gives A372517, cf. A014499, A061712.
For squarefree instead of prime we have A372683, cf. A143658, A372540.
The indices of these prime are given by A372684.
Cf. A007918, A007920, A029837, A035100, A049095, A130739.

Join[{2,2},NextPrime[#]&/@(2^Range[2,40])]  (* Harvey P. Dale, Jan 26 2011 *)
NextPrime[2^Range[0,50]-1]  (* Vladimir Joseph Stephan Orlovsky, Apr 11 2011 *)

g(n,b=2) = for(x=0,n,print1(nextprime(b^x)","))

a(n) = nextprime(2^n); \\ Michel Marcus, Nov 01 2020

a(n) = A014210(n), n <> 1. - R. J. Mathar, Oct 14 2008
Sum_{n >= 0} 1/a(n) = A338475 + 1/6 = 1.4070738... (because 1/6 = 1/2 - 1/3). - Bernard Schott, Nov 01 2020
From Gus Wiseman, Jun 03 2024: (Start)
a(n) = A007918(2^n).
a(n) = 2^n + A092131(n).
a(n) = prime(A372684(n)).
(End)

A104082 Smallest prime >= 4^n.

Original entry on oeis.org

2, 5, 17, 67, 257, 1031, 4099, 16411, 65537, 262147, 1048583, 4194319, 16777259, 67108879, 268435459, 1073741827, 4294967311, 17179869209, 68719476767, 274877906951, 1099511627791, 4398046511119, 17592186044423, 70368744177679, 281474976710677, 1125899906842679

Offset: 0

Cino Hilliard, Mar 03 2005

Vincenzo Librandi, Table of n, a(n) for n = 0..500

Cf. A104080 (for 2^n), A104081 (for 3^n).

Cf. A014210.

[NextPrime(4^n-1): n in [0..50]]; // Jinyuan Wang, Nov 09 2018

NextPrime[4^Range[0, 23]] (* Vladimir Joseph Stephan Orlovsky, Mar 13 2011 *)

g(n,b=4) = for(x=0,n,print1(nextprime(b^x)","))

a(n) = A104080(2n). - Jinyuan Wang, Nov 09 2018

A340959 Table read by antidiagonals of the smallest prime >= n^k, n >= 1 and k >= 0.

Original entry on oeis.org

2, 2, 2, 2, 2, 2, 2, 3, 5, 2, 2, 5, 11, 11, 2, 2, 5, 17, 29, 17, 2, 2, 7, 29, 67, 83, 37, 2, 2, 7, 37, 127, 257, 251, 67, 2, 2, 11, 53, 223, 631, 1031, 733, 131, 2, 2, 11, 67, 347, 1297, 3137, 4099, 2203, 257, 2, 2, 11, 83, 521, 2411, 7789, 15629, 16411, 6563

Offset: 1

Donald S. McDonald, Jan 31 2021

			Table begins:
  2, 2,  2,   2,   2,    2, ...
  2, 2,  5,  11,  17,   37, ...
  2, 3, 11,  29,  83,  251, ...
  2, 5, 17,  67, 257, 1031, ...
  2, 5, 29, 127, 631, 3137, ...
  ...;
yielding the triangle:
  2;
  2, 2;
  2, 2,  2;
  2, 3,  5,  2;
  2, 5, 11, 11,  2;
  2, 5, 17, 29, 17, 2;
  ...

Cf. A104080 (n=2), A104081 (n=3), A104082 (n=4), A104083 (n=5), A104084 (n=7).

T[n_,k_]:=NextPrime[n^k-1];Flatten[Table[T[n-k,k],{n,11},{k,0,n-1}]] (* Stefano Spezia, Feb 01 2021 *)

T(n,k) = nextprime(n^k); \\ Michel Marcus, Feb 01 2021

T(n,k) = next_prime(n^k-1).

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A104080 Smallest prime >= 2^n.

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A104082 Smallest prime >= 4^n.

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A340959 Table read by antidiagonals of the smallest prime >= n^k, n >= 1 and k >= 0.

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Author

Keywords

Examples

Crossrefs

Programs

Mathematica

PARI

Formula