A051254 -id:A051254 - cp's OEIS frontend

A123561 Continued fraction expansion of Mills' constant.

1, 3, 3, 1, 3, 1, 2, 1, 2, 1, 4, 2, 35, 21, 1, 4, 4, 1, 1, 3, 2, 17, 7, 4, 1, 3, 16, 5, 3, 2, 3, 1, 4, 8, 1, 1, 19578, 1, 1, 1, 1, 1, 8, 1, 2, 1, 1, 3, 2, 1, 1, 1, 2, 1, 1, 9, 1, 1, 1, 2, 14, 4, 19, 1, 3, 3, 5, 9, 1, 3, 1, 6, 1, 2, 1, 11, 1, 1, 5, 23, 1, 4, 2, 3, 1, 35, 2, 1, 5, 3, 1, 9, 4, 8, 3, 1, 6, 2, 1

Offset: 0

Hauke Worpel (hw1(AT)email.com), Nov 11 2006

G. C. Greubel, Table of n, a(n) for n = 0..9999
Eric Weisstein's World of Mathematics, Mills' constant

Cf. A051021 (decimal expansion), A051254.

ContinuedFraction[Nest[NextPrime[#^3] &, 2, 7]^(1/3^8), 100] (* G. C. Greubel, Oct 25 2017 *)

More terms from Robert G. Wilson v, Nov 18 2006

Offset changed by Andrew Howroyd, Aug 09 2024

A171883 Mills primes, starting with 3.

Original entry on oeis.org

3, 29, 24391, 14510715208481, 3055388613462301256452407743005777548691

Offset: 1

Robert Munafo, Feb 27 2010

nonn

For the standard Mills primes sequence, A051254, one starts with 2, and each successive term a(n) is the smallest prime greater than a(n-1)^3. This sequence uses the same definition but starts with 3.

a(6) has 119 digits and is too large to include.

Vincenzo Librandi, Table of n, a(n) for n = 1..7
Robert Munafo, Mills primes, starting with 3

Cf. A051254

p = 36/25; Table[p = NextPrime[p^3], {6}] (* From Alonso del Arte based on T. D. Noe's program for A051254, Oct 05 2011 *)
NestList[NextPrime[#^3]&,3,5] (* Harvey P. Dale, Feb 15 2014 *)

n:3 $ l:10^100 $ print(n) $ while (n

Offset corrected by Arkadiusz Wesolowski, Oct 05 2011

A300753 Decimal expansion of the constant B such that ceiling(B^(3^k)) = A118910(k) is prime for all k >= 0.

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, Jun 19 2018

Tóth calculated the first 5500 decimal digits of this constant. The first 600 digits are presented in his paper.

			1.24055470525201424067469515337900345212353396725255...

László Tóth, A Variation on Mills-Like Prime-Representing Functions, Journal of Integer Sequences, Vol. 20 (2017), Article 17.9.8.

Cf. A051021, A051254, A118910.

Lim_{n->oo} (A118910(n) - 1)^(3^(-n)).

A382261 a(n) = floor(x^(phi^n)), where phi = (1+sqrt(5))/2 and x is the constant A382260.

Original entry on oeis.org

2, 3, 7, 23, 163, 3803, 620549, 2359981439, 1464484123012601, 3456155348019933976288373, 5061484633840283809323162088349619180781, 17493277186167814180104995425523045477935447066389138909089293633

Offset: 1

Thomas Scheuerle, Mar 19 2025

nonn

Conjecture: All terms are prime numbers. For details see A382260.

Cf. A001622, A059784, A051254, A243358, A382260. A090253.

Cf. A090253 ( similar growth ).

nextprime(a(n-2)*a(n-1)) <= a(n) < nextprime((a(n-2)+1)*a(n-1)).

A286682 a(n) = A059784(n+1) - A059784(n)^2.

Original entry on oeis.org

1, 4, 12, 4, 22, 12, 114, 4, 138, 142, 2956, 6388, 5248, 17532, 96930, 83782, 1464, 897448, 300832, 26908

Offset: 1

Jeppe Stig Nielsen, May 12 2017

This sequence relates to A059784 just like A108739 relates to the Mills primes A051254.
That this leads to a Mills-like real constant C such that floor(C^2^n) is a prime number for any natural number n, requires a proof of Legendre's conjecture that there is always a prime between consecutive perfect squares.
a(18) and a(19) generate 96042- and 192083-decimal digit probable primes. - Serge Batalov, May 27 2024
a(20) generates a 384166-decimal digit probable prime. - Serge Batalov, May 27 2024

			A059784(8) by construction can be written ((((((2^2 + 1)^2 + 4)^2 + 12)^2 + 4)^2 + 22)^2 + 12)^2 + 114. Taking out the addends gives 1, 4, 12, 4, 22, 12, 114 which lists the first seven terms of this sequence.

Cf. A059784, A108739, A051254.

Map[#2 - #1^2 & @@ # &, Partition[NestList[NextPrime[#^2] &, 2, 12], 2, 1]] (* Michael De Vlieger, May 12 2017 *)

p=2;while(1,a=nextprime(p^2);print1(a-p^2,", ");p=a)

a(14)-a(17) from Serge Batalov, May 26 2024

a(18)-a(20) from Serge Batalov, May 27 2024

A323828 Decimal expansion of a constant related to Bertrand's Postulate.

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane, Feb 01 2019

			3.5679760909846634612366007737192526425527943028977913311170116688989193092...

Dylan Fridman et al., A prime-representing constant, Amer. Math. Monthly 126 (2019), 72-73 (on ResearchGate).

Cf. A249720, A051254.

evalf(Sum((2^(k-1) + 1)/Product(2^(i-1) + 2, i=1..k-1), k=1..infinity), 120); # Vaclav Kotesovec, Jun 04 2019

def g(n):
    return sum((2^(k-1) + 1)/prod(2^(i-1) + 2 for i in (1..k-1))
               for k in (1..n))
N(g(100), digits=102) # Freddy Barrera, Feb 17 2019

Sum_{k>=1} (2^(k-1) + 1)/(Product_{i=1..k-1} 2^(i-1) + 2). See Dylan Fridman et al. reference. - Freddy Barrera, Feb 17 2019

More terms from Freddy Barrera, Feb 17 2019

cp's OEIS Frontend

A123561 Continued fraction expansion of Mills' constant.

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A171883 Mills primes, starting with 3.

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A300753 Decimal expansion of the constant B such that ceiling(B^(3^k)) = A118910(k) is prime for all k >= 0.

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A382261 a(n) = floor(x^(phi^n)), where phi = (1+sqrt(5))/2 and x is the constant A382260.

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A286682 a(n) = A059784(n+1) - A059784(n)^2.

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A323828 Decimal expansion of a constant related to Bertrand's Postulate.

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