A112597 -id:A112597 - cp's OEIS frontend

A059784 a(n+1) = nextprime(a(n)^2). Smallest prime following the square of previous prime. Initial value = 2.

2, 5, 29, 853, 727613, 529420677791, 280286254072681840639693, 78560384222095957698731679318817728959447134363

Offset: 1

Labos Elemer, Feb 22 2001

nonn

Charles R Greathouse IV, Table of n, a(n) for n = 1..12
Kaisa Matomäki, Prime-representing functions, Acta Math. Hungar. 128 (2010), pp. 307-314.

Cf. A006992, A055496, A005384, A005385, A112597.

p=2; lst={p}; Do[p=NextPrime[p*p]; AppendTo[lst,p], {n,8}]; lst (* Vladimir Joseph Stephan Orlovsky, May 27 2009 *)
NestList[NextPrime[#^2] &, 2, 7] (* Michael De Vlieger, Nov 02 2017 *)

a(n) = floor[1.5246999605380943599233635756884211622202236231...^(2^n)], similar to Mills Primes A051254. - Henry Bottomley, Oct 19 2003

Changed offset to 1 to parallel other such sequences. - Robert G. Wilson v, Nov 15 2012

A219177 Decimal expansion of what appears to be the smallest possible C for which the nearest integer to C^2^n is always prime and starts with 2.

Original entry on oeis.org

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

Offset: 1

Robert G. Wilson v, Nov 15 2012

The square of this constant, C^2 = 1.6180339472264..., is very close to the Golden Ratio Phi (A001622).
This constant is about 3% less than Mills's constant, 1.306377883863..., (A051021).
Since there is always a prime between an integer and its square, this constant should satisfy the same criteria as does Mills's constant (A051021).
This constant, C, produces A059785.

			=1.2720196331921934958697353209119288376375630826996476481322580415...

Robert G. Wilson v, Table of n, a(n) for n = 1..1000

Cf. A051021, A059785, A112597, A117739.

RealDigits[ Nest[ NextPrime[#^2, -1] &, 2, 8]^(2^-9), 10, 111][[1]]

A382260 Decimal expansion of x, where x is the smallest number for which floor(x^(phi^k)) is prime for k > 0 where phi = (1+sqrt(5))/2, assuming that Oppermann's conjecture holds.

Original entry on oeis.org

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

Offset: 1

Thomas Scheuerle, Mar 19 2025

This constant can generate for all exponents k > 0 a prime number if the following conjecture is true: Let p be a prime > 2 and q = nexprime(p+1) then if there is always at least one prime inside the interval nextprime(p*q) to nextprime((p+1)*q)). But if this constant can generate prime numbers for all k, it is not directly a proof of this conjecture. If we would strengthen this further by omitting "nextprime" and allowing natural numbers for p and q, we will obtain essentially Oppermann's conjecture.

			1.5831204048581...

Wikipedia, Oppermann's conjecture.

Cf. A001622, A051021, A112597, A382261.

floor(x^(phi^n)) = A382261(n) where x is this constant.

cp's OEIS Frontend

A059784 a(n+1) = nextprime(a(n)^2). Smallest prime following the square of previous prime. Initial value = 2.

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A219177 Decimal expansion of what appears to be the smallest possible C for which the nearest integer to C^2^n is always prime and starts with 2.

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Programs

Mathematica

A382260 Decimal expansion of x, where x is the smallest number for which floor(x^(phi^k)) is prime for k > 0 where phi = (1+sqrt(5))/2, assuming that Oppermann's conjecture holds.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula