A144256 -id:A144256 - cp's OEIS frontend

A143935 Number of primes between n^K and (n+1)^K, inclusive, where K=log(127)/log(16).

2, 1, 2, 1, 2, 2, 2, 2, 2, 2, 3, 2, 2, 5, 1, 4, 2, 3, 3, 4, 1, 5, 3, 3, 4, 3, 3, 3, 4, 4, 3, 5, 4, 3, 5, 2, 4, 5, 4, 5, 5, 3, 5, 5, 2, 6, 5, 4, 4, 4, 5, 5, 7, 5, 5, 3, 5, 6, 3, 8, 3, 4, 5, 6, 7, 5, 6, 8, 5, 4, 6, 6, 3, 7, 5, 4, 8, 5, 8, 6, 3, 7, 7, 6, 8, 7, 4, 5, 6, 5, 9, 9, 7, 6, 6, 6, 6, 7, 6, 4, 8, 5, 8, 8, 4

Offset: 1

T. D. Noe, Sep 05 2008

nonn

This value of K is conjectured to be the least possible such that there is at least one prime in the range n^k and (n+1)^k for all n>0 and k>=K. This value of K was found using exact interval arithmetic. For each n <= 300 and for each prime p in the range n to n^2, we computed an interval k(n,p) such that p is between n^k(n,p) and (n+1)^k(n,p). The intersection of all these intervals produces a list of 29 intervals. The last interval appears to be semi-infinite beginning with K, which is log(127)/log(16). See A143898 for the smallest number in the first interval.
My UBASIC program indicates no prime between 113.457 ... and 126.999 .... Next prime > 113 is 127. I would like someone to check this. - Enoch Haga, Sep 24 2008
It suffices to check members of floor(A002386^(1/k)). - Charles R Greathouse IV, Feb 03 2011
The constant log(127)/log(16) is A194361. - John W. Nicholson, Dec 13 2013

T. D. Noe, Table of n, a(n) for n = 1..10000
Carlos Rivera, Conjecture 60: Generalization of Legendre's Conjecture

Cf. A014085 (number of primes between n^2 and (n+1)^2).

Cf. A144256, A194361.

k= 1.74717117169304146332; Table[Length[Select[Range[Ceiling[n^k],Floor[(n+1)^k]], PrimeQ]], {n,150}]
With[{k=Log[16,127]},Table[Count[Range[Ceiling[n^k],Floor[(n+1)^k]],?PrimeQ],{n,110}]] (* _Harvey P. Dale, Apr 03 2019 *)

Corrected a(15) from 1 to 0 Enoch Haga, Sep 24 2008

My intention was to include the endpoints of the range. Using k=log(127)/log(16), the endpoint for n=15 is exactly 127, which is prime. - T. D. Noe, Sep 25 2008

A144831 (n+1)^2 - (smallest prime > n^2).

Original entry on oeis.org

2, 4, 5, 8, 7, 12, 11, 14, 17, 20, 17, 20, 23, 28, 29, 32, 31, 30, 33, 40, 41, 42, 35, 48, 45, 52, 51, 54, 47, 54, 57, 58, 65, 62, 67, 72, 71, 74, 77, 80, 71, 72, 75, 76, 89, 80, 91, 92, 89, 98, 95, 102, 97, 108, 99, 112, 113, 110, 109, 114, 117, 122, 107, 126, 127, 132, 131

Offset: 1

Enoch Haga, Sep 21 2008

Suggested by Conjecture 60 in Carlos Rivera's The Prime Puzzles & Problems Connection.

Legendre's conjecture that there is always a prime between n^2 and (n+1)^2 is equivalent to a(n) >= 0 for all n. As the conjecture is still opened, it is not proved that a(n) is nonn, although the keyword is automatically added. - Jean-Christophe Hervé, Oct 26 2013

			a(2)=4 because n=2, 2^2=4 and (2+1)^2=9. The gap in which primes are to be found is 4 - 9. Next prime=5 and 9-5=4. For a(3)=5, 3^2=9 and (3+1)^2=16. Next prime=11 and 16-11=5.

Jean-Christophe Hervé, Table of n, a(n) for n = 1..10000

Cf. A007491, A053000, A144832, A144256, A014085.

Table[n^2-NextPrime[(n-1)^2],{n,2,70}] (* Harvey P. Dale, Jan 22 2019 *)

a(n) = (n+1)^2 - nextprime(n^2); \\ Michel Marcus, Jun 08 2014

Calculate n^2 and (n+1)^2, e.g. 4 - 9. Find the next prime following n^2 and subtract from (n+1)^2. Next prime is 5 so 9-5=4, the distance from next prime to (n+1)^2.

a(n) = (n+1)^2 - A007491(n).

Definition rewritten by N. J. A. Sloane, Sep 28 2008

Definition rewritten by Jean-Christophe Hervé, Oct 26 2013

A194361 Decimal expansion of log(127) / log(16).

Original entry on oeis.org

1, 7, 4, 7, 1, 7, 1, 1, 7, 1, 6, 9, 3, 0, 4, 1, 4, 6, 3, 3, 2, 0, 0, 0, 0, 9, 7, 3, 0, 7, 5, 4, 2, 5, 6, 9, 2, 4, 2, 0, 3, 9, 1, 1, 6, 1, 7, 3, 7, 7, 9, 6, 6, 3, 3, 2, 3, 6, 0, 2, 5, 5, 1, 1, 7, 8, 1, 4, 5, 6, 5, 7, 6, 2, 2, 0, 5, 0, 4, 1, 0, 3, 2, 2, 7, 0, 1, 1, 4, 5, 1, 3, 0, 9, 6, 6, 9, 1, 9, 6, 5, 1, 0, 3, 2

Offset: 1

T. D. Noe, Oct 04 2011

			1.747171171693041463320000973...

G. C. Greubel, Table of n, a(n) for n = 1..5000
Index entries for transcendental numbers

Cf. A143935, A144256.

Mathematica
```
RealDigits[Log[127]/Log[16], 10, 105]
```

log(127)/log(16) \\ G. C. Greubel, Feb 19 2017

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A143935 Number of primes between n^K and (n+1)^K, inclusive, where K=log(127)/log(16).

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A144831 (n+1)^2 - (smallest prime > n^2).

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A194361 Decimal expansion of log(127) / log(16).

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