A144831 -id:A144831 - cp's OEIS frontend

A144832 Distance from nxtprm(n^2) to (n+1)^2 in A144831 is prime.

2, 5, 7, 11, 17, 17, 23, 29, 31, 41, 47, 67, 71, 71, 89, 89, 97, 113, 109, 107, 127, 131, 137, 157, 167, 173, 173, 191, 197, 193, 197, 227, 233, 227, 251, 257, 271, 293, 271, 307, 313, 317, 331, 349, 353, 383, 383, 409, 419, 431, 449, 463, 467, 487, 503, 509

Offset: 1

Enoch Haga, Sep 22 2008

			a(2)=5 because 3^2=9 and 4^2=16. Nxtprm(3^2)=11 and 16-11=5, a prime.

A007491 A053000 A144831

Terms in this sequence are from A144831 iff the distance from nxtprm n^2 to (n+1)^2 is prime.

A007491 Smallest prime > n^2.

Original entry on oeis.org

2, 5, 11, 17, 29, 37, 53, 67, 83, 101, 127, 149, 173, 197, 227, 257, 293, 331, 367, 401, 443, 487, 541, 577, 631, 677, 733, 787, 853, 907, 967, 1031, 1091, 1163, 1229, 1297, 1373, 1447, 1523, 1601, 1693, 1777, 1861, 1949, 2027, 2129, 2213, 2309, 2411, 2503

Offset: 1

N. J. A. Sloane, Robert G. Wilson v, R. K. Guy

Suggested by Legendre's conjecture (still open) that there is always a prime between n^2 and (n+1)^2.
Legendre's conjecture is equivalent to a(n) < (n+1)^2. - Jean-Christophe Hervé, Oct 26 2013
From Jaroslav Krizek, Apr 02 2016: (Start)
Conjectures:
1) There is always a prime p between n^2 and n^2+n (verified up to 13*10^6).
2) a(n) is the smallest prime p such that n^2 < p < n^2+n; a(n) < n^2+n.
3) For all numbers k >= 1 there is the smallest number m > 2*(k+1) such that for all numbers n >= m there is always a prime p between n^2 and n^2 + n - 2k. Sequence of numbers m for k >= 1: 6, 8, 12, 13, 14, 24, 24, 24, 30, 30, 30, 31, 33, 35, 43, ...; lim_{k->oo} m/2k = 1. Example: k=2; for all numbers n >= 8 there is always a prime p between n^2 and n^2 + n - 4. (End)

Archimedeans Problems Drive, Eureka, 24 (1961), 20.
J. R. Goldman, The Queen of Mathematics, 1998, p. 82.
G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. 3rd ed., Oxford Univ. Press, 1954, p. 19.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Jean-Christophe Hervé, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)
Eric Weisstein's World of Mathematics, Landau's Problem.
Eric Weisstein's World of Mathematics, Legendre's Conjecture.

Cf. A053000, A053001, A014085, A144831.

Cf. A007918, A000290.

a007491 = a007918 . a000290  -- Reinhard Zumkeller, Jun 07 2015

[NextPrime(n^2): n in [1..50]]; // Vincenzo Librandi, Apr 30 2015

Maple
```
[seq(nextprime(i^2), i=1..100)];
```

NextPrime[Range[60]^2]  (* Harvey P. Dale, Mar 24 2011 *)

PARI
```
vector(100,i,nextprime(i^2))
```

from sympy import nextprime
def a(n): return nextprime(n**2)
print([a(n) for n in range(1, 51)]) # Michael S. Branicky, Jan 13 2023

a(n) = A007918(A000290(n)). - Reinhard Zumkeller, Jun 07 2015

More terms from Labos Elemer, Nov 17 2000

Definition modified by Jean-Christophe Hervé, Oct 26 2013

A144256 a(n) = smallest prime in the range [n^K, (n+1)^K], where K = log(127)/log(16) = 1.74717117169304146332...

Original entry on oeis.org

2, 5, 7, 13, 17, 23, 31, 41, 47, 59, 67, 79, 89, 101, 127, 127, 149, 157, 173, 191, 211, 223, 241, 263, 277, 307, 317, 347, 359, 383, 409, 431, 457, 479, 499, 541, 557, 577, 607, 631, 659, 691, 719, 751, 787, 809, 839, 877, 907, 937, 967, 997, 1031, 1069

Offset: 1

Alexander R. Povolotsky, Sep 16 2008

nonn

T. D. Noe submitted to primepuzzles.net the following conjecture #60, which is stronger than the Legendre's conjecture: For n>0 and k>=K, there is always a prime between n^k and (n+1)^k, where K = log(127)/log(16) = 1.74717117169304146332...
One could see that calculated terms for n=15 and n=16 yield the same value: 127, which make this conjecture (as originally defined) to be questionable. If this conjecture is modified to k>K, then there will be a distinct prime between 15^k and 16^k. It appears that the relatively large prime gap between 113 and 127 is the largest gap to overcome. Another way to correct/clarify the conjecture is to mention that both boundaries of the interval are included and that the same prime value may appear in two neighboring intervals. Of course the last version of the modified definition makes this conjecture to be different from the original Legendere conjecture (rather than to be an improvement of the original Legendere conjecture). [Alexander R. Povolotsky, Sep 26 2008]
The constant log(127)/log(16) is A194361. - John W. Nicholson, Dec 13 2013

G. C. Greubel, Table of n, a(n) for n = 1..5000
Carlos Rivera, Conjecture 60: Generalization of Legendre's Conjecture

Cf. A000040, A144831, A143935.

k = Log[127]/Log[16]; Table[Select[Range[Ceiling[n^k], Floor[(n + 1)^k]], PrimeQ, 1][[1]], {n, 100}] (* T. D. Noe, Jan 30 2012 *)

i=[];for(n=1, 500, for(j=ceil(n^(log(127)/log(16))), ceil((n+1)^(log(127)/log(16))), if(isprime(j), i=concat(i,j)); if(isprime(j), break)));i

Extended by T. D. Noe, Jan 30 2012

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A144832 Distance from nxtprm(n^2) to (n+1)^2 in A144831 is prime.

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A007491 Smallest prime > n^2.

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A144256 a(n) = smallest prime in the range [n^K, (n+1)^K], where K = log(127)/log(16) = 1.74717117169304146332...

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Extensions