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.

A014210 Next prime after 2^n.

Original entry on oeis.org

2, 3, 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

Views

Author

N. J. A. Sloane

Keywords

Comments

Except for a(1) = 3 instead of 2, a(n) is the least prime obtained as a binomial transform of n numbers. E.g. a(5) = (1,5,10,10,5,1).(1,1,1,1,1,6)= 37. - Amarnath Murthy, Nov 26 2003
a(n) is the smallest m for which m>(tau(m))^n, where tau(m) is the number of divisors of m. [Vladimir Shevelev, May 31 2010]
Equivalently, "Smallest prime > 2^n" while in A104080 it is "Smallest prime >= 2^n". The only difference is the 2nd term with a(1) = 3 and A104080(1) = 2. - Bernard Schott, Oct 30 2020

References

  • J.-M. De Koninck & A. Mercier, 1001 Problèmes en Théorie Classique des Nombres, Problème 615 pp. 82 and 279, Ellipses, Paris, 2004. Warning : gives Sum_{k>=1} 1/A104080(k) = 0.7404...

Links

Crossrefs

See A203074 for another version.
Cf. A014211, A014234, A054321, A104080, A338475.

Programs

  • Maple
    [ seq( nextprime( 2^i ),i=0..40) ];
  • Mathematica
    NextPrime[ n_Integer] := (k = n + 1; While[ !PrimeQ[k], k++ ]; k); Table[ NextPrime[2^n], {n, 0, 35} ]
f[n_] := NextPrime[2^n]; Array[f, 30, 0] (* Robert G. Wilson v, Jun 05 2015 *)
NextPrime[2^Range[0,40]] (* Harvey P. Dale, Jun 22 2017 *)
  • PARI
    a(n) = nextprime(2^n+1); \\ Michel Marcus, Oct 30 2020

Formula

Sum_{k>=0} 1/a(k) = A338475. - Bernard Schott, Oct 30 2020

A074496 a(n) = smallest prime > e^n.

Original entry on oeis.org

2, 3, 11, 23, 59, 149, 409, 1097, 2999, 8111, 22027, 59879, 162779, 442439, 1202609, 3269029, 8886113, 24154957, 65660003, 178482319, 485165237, 1318815761, 3584912873, 9744803489, 26489122147, 72004899361, 195729609461, 532048240609, 1446257064299, 3931334297161
Offset: 0

Views

Author

Joseph L. Pe, Sep 26 2002

Keywords

Examples

			The first prime > e^3 = 20.085... is 23, so a(3) = 23.

Links

Crossrefs

Cf. A000040, A000149, A001113, A014210, A014211, A074497.

Programs

  • Mathematica
    a[n_] := NextPrime[Exp[n]]; a /@ Range[0, 20] (* Giovanni Resta, Apr 03 2017 *)
  • PARI
    for(n=1,50,print1(nextprime(exp(n))","))

Formula

Limmit_{n -> infinity} a(n+1)/a(n) = e. - Jonathan Vos Post, Apr 30 2006

Extensions

More terms from Ralf Stephan, Mar 25 2003
Edited by N. J. A. Sloane, Dec 22 2006
a(18) inserted and more terms added by Amiram Eldar, Sep 30 2019

A054321 Smallest prime greater than 5^n.

Original entry on oeis.org

2, 7, 29, 127, 631, 3137, 15629, 78137, 390647, 1953151, 9765629, 48828139, 244140683, 1220703131, 6103515637, 30517578167, 152587890649, 762939453127, 3814697265637, 19073486328181, 95367431640673, 476837158203149
Offset: 0

Views

Author

Robert G. Wilson v, Aug 14 2001

Keywords

Links

Crossrefs

Cf. A014210, A014211, A013599 (a(n)-5^n).

Programs

  • Maple
    seq(nextprime(5^n),n=0..100); # Robert Israel, May 19 2014
  • Mathematica
    NextPrime[ n_Integer] := (k = n + 1; While[ !PrimeQ[k], k++ ]; k); Table[ NextPrime[5^n], {n, 0, 22} ] (* Mathematica 5 and below *)
NextPrime[5^Range[0,25]] (* Mathematica 6; Harvey P. Dale, Jun 19 2011 *)
  • PARI
    a(n)=nextprime(5^n+1) \\ Charles R Greathouse IV, Jun 19 2011

A104081 Smallest prime >= 3^n.

Original entry on oeis.org

2, 3, 11, 29, 83, 251, 733, 2203, 6563, 19687, 59051, 177167, 531457, 1594331, 4782971, 14348909, 43046747, 129140197, 387420499, 1162261523, 3486784409, 10460353259, 31381059613, 94143178859, 282429536483, 847288609457
Offset: 0

Views

Author

Cino Hilliard, Mar 03 2005

Keywords

Links

Crossrefs

Cf. A104080 (for 2^n).
Cf. A104088 (largest prime <= 3^n).
Cf. A000244, A007918, A014211.

Programs

Formula

a(n) = A014211(n), n > 1. - R. J. Mathar, Dec 13 2008
a(n) = A007918(A000244(n)). - Michel Marcus, Nov 08 2018

A074497 a(n) = the smallest prime > Pi^n.

Original entry on oeis.org

2, 5, 11, 37, 101, 307, 967, 3023, 9491, 29819, 93683, 294211, 924281, 2903689, 9122207, 28658153, 90032231, 282844571, 888582407, 2791563953, 8769956803, 27551631847, 86556004201, 271923706939, 854273519929, 2683779414353, 8431341691901, 26487841119121
Offset: 0

Views

Author

Joseph L. Pe, Sep 26 2002

Keywords

Comments

The corresponding indices in A000040 are the sequence {1, 3, 5, 12, 26, 63, 163, 434, 1176, 3230, 9043, 25550, 73051, ...}. - L. Edson Jeffery, Jan 17 2014

Examples

			The first prime > Pi^3 = 31.006.... is 37, so a(3) = 37.

Crossrefs

Cf. A000040, A014210, A014211, A074496.

Programs

  • Mathematica
    a[n_] := NextPrime[Pi^n];  a /@ Range[0,10] (* Giovanni Resta, Jan 17 2014 *)
  • PARI
    with \p1000: for(n=0,30,print1(nextprime(Pi^n)","))

Extensions

More terms from Ralf Stephan, Mar 19 2003

A037128 a(n) = nextprime(3^n) - nextprime(2^n).

Original entry on oeis.org

0, 2, 6, 18, 66, 214, 666, 2072, 6306, 19166, 58020, 175114, 527358, 1586122, 4766560, 14316138, 42981210, 129009096, 387158352, 1161737214, 3485735826, 10458256090, 31376865294, 94134790242, 282412759224, 847255054990, 2541798719452, 7625463267246, 22876524019528, 68629840494090, 205890058352826
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Crossrefs

Cf. A014210, A014211.

Programs

  • Mathematica
    Table[NextPrime[3^n] - NextPrime[2^n], {n, 0, 30}] (* Vincenzo Librandi, Jan 22 2015 *)
  • PARI
    a(n) = nextprime(3^n+1)-nextprime(2^n+1); \\ Michel Marcus, Jan 22 2015

Formula

a(n) = A014211(n) - A014210(n). - Michel Marcus, Jan 22 2015

Extensions

More terms from Vincenzo Librandi, Jan 22 2015
Showing 1-6 of 6 results.

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

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