A108739 -id:A108739 - cp's OEIS frontend

A051254 Mills primes.

2, 11, 1361, 2521008887, 16022236204009818131831320183, 4113101149215104800030529537915953170486139623539759933135949994882770404074832568499

Offset: 1

Simon Plouffe

Mills showed that there is a number A > 1 but not an integer, such that floor( A^(3^n) ) is a prime for all n = 1, 2, 3, ... A is approximately 1.306377883863... (see A051021).
a(1) = 2 and (for n > 1) a(n) is least prime > a(n-1)^3. - Jonathan Vos Post, May 05 2006, corrected by M. F. Hasler, Sep 11 2024
The name refers to the American mathematician William Harold Mills (1921-2007). - Amiram Eldar, Jun 23 2021

			a(3) = 1361 = 11^3 + 30 = a(2)^3 + 30 and there is no smaller k such that a(2)^3 + k is prime. - _Jonathan Vos Post_, May 05 2006

Tom M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 8.
Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 2.13, p. 130.
Paulo Ribenboim, The Little Book of Bigger Primes, Springer-Verlag NY 2004. See p. 137.

Robert G. Wilson v, Table of n, a(n) for n = 1..8
Chris K. Caldwell, Mills' Theorem - a generalization.
Chris K. Caldwell and Yuanyou Chen, Determining Mills' Constant and a Note on Honaker's Problem, Journal of Integer Sequences, Vol. 8 (2005), Article 05.4.1.
Steven R. Finch, Mills' Constant. [Broken link]
Steven R. Finch, Mills' Constant. [From the Wayback machine]
Dylan Fridman, Juli Garbulsky, Bruno Glecer, James Grime and Massi Tron Florentin, A Prime-Representing Constant, Amer. Math. Monthly, Vol. 126, No. 1 (2019), pp. 72-73; ResearchGate link, arXiv preprint, arXiv:2010.15882 [math.NT], 2020.
James Grime and Brady Haran, Awesome Prime Number Constant, Numberphile video, 2013.
Brian Hayes, Pumping the Primes, bit-player, Aug 19 2015.
Ernest G. Hibbs, Component Interactions of the Prime Numbers, Ph. D. Thesis, Capitol Technology Univ. (2022), see p. 33.
William H. Mills, A prime-representing function, Bull. Amer. Math. Soc., Vol. 53, No. 6 (1947), p. 604; Errata, ibid., Vol. 53, No 12 (1947), p. 1196.
Simon Plouffe, The calculation of p(n) and pi(n), arXiv:2002.12137 [math.NT], 2020.
László Tóth, A Variation on Mills-Like Prime-Representing Functions, arXiv:1801.08014 [math.NT], 2018.
Juan L. Varona, A Couple of Transcendental Prime-Representing Constants, arXiv:2012.11750 [math.NT], 2020.
Eric Weisstein's World of Mathematics, Mills' Prime.
Eric Weisstein's World of Mathematics, Prime Formulas.
Eric W. Weisstein, Table of n, a(n) for n = 1..13.

Cf. A001358, A055496, A076656, A006992, A005384, A005385, A118908, A118909, A118910, A118911, A118912, A118913.
Cf. A224845 (integer lengths of Mills primes).
Cf. A108739 (sequence of offsets b_n associated with Mills primes).
Cf. A051021 (decimal expansion of Mills constant).

floor(A^(3^n), n=1..10); # A is Mills's constant: 1.306377883863080690468614492602605712916784585156713644368053759966434.. (A051021).

p = 1; Table[p = NextPrime[p^3], {6}] (* T. D. Noe, Sep 24 2008 *)
NestList[NextPrime[#^3] &, 2, 5] (* Harvey P. Dale, Feb 28 2012 *)

a(n)=if(n==1, 2, nextprime(a(n-1)^3)) \\ Charles R Greathouse IV, Jun 23 2017

apply( {A051254(n, p=2)=while(n--, p=nextprime(p^3));p}, [1..6]) \\ M. F. Hasler, Sep 11 2024

a(1) = 2; a(n) is least prime > a(n-1)^3. - Jonathan Vos Post, May 05 2006

Edited by N. J. A. Sloane, May 05 2007

A060449 Generalized Mills numbers: a(n) = floor(c^(b^n)) where c=4.4, b=1.179.

Original entry on oeis.org

5, 7, 11, 17, 29, 53, 109, 252, 679, 2184, 8650, 43828, 296913, 2832896, 40474353, 930818760, 37522518949, 2931502379404, 499688559138590, 213681340556825047, 270268120176240462240, 1227682581046707804164120

Offset: 1

Jason Earls, Apr 07 2001

First seven terms are primes.

Harry J. Smith, Table of n, a(n) for n = 1..44
C. Rivera, Prime Puzzles

Cf. A051254, A108739, A051021, A060699, A191357.

Digits := 100; A060449 := n->4.4^(1.179^n);

{ default(realprecision, 2000); for (n=1, 44, write("b060449.txt", n, " ", floor(4.4^(1.179^n))); ) } \\ Harry J. Smith, Jul 05 2009

More terms from James Sellers, Apr 11 2001

Offset changed from 0 to 1 by Harry J. Smith, Jul 05 2009

A060699 a(n) = floor(A^(C^n)), where A = 2.084551112207285611..., C = 1.221.

Original entry on oeis.org

2, 2, 3, 5, 7, 11, 19, 37, 83, 223, 739, 3181, 18911, 166679, 2376391, 60953117, 3202432763, 403823050201

Offset: 1

Luis Rodriguez-Torres (ludovicusmagister(AT)yahoo.com), Apr 20 2001

nonn

Results from the application of Caldwell's Generalized Mills's Theorem. This value of A produces 18 primes. For 20 primes A must be adjusted to 2.084551112207285611.

The extension of the sequence is guaranteed by the Cramer conjecture. That is: If the needed change in Y(n) for obtaining the next prime (superior or inferior) is as maximum = (log Y(n))^2/2, then the effect on Y(n-1) is less than K*C^(2n-1)*Y(n-1)/Y(n). K = (1/2)*(log A)^2 = 0.269784 This value diminishes with n. Example: For n = 23, a change in Y(23) by 2630 only changes Y(22) by 0.0043. Jens Kruse Anderson with A = 2.084551112197624209091521123 calculated Y(n) = floor(A^(C^n)) from n = 1 to n = 3, obtaining 22 different primes. - Luis Rodriguez-Torres (ludovicusmagister(AT)yahoo.com), Feb 10 2009

			a(10) = 223 because 2.0845511122073^(1.221^10)= 223.58376...
With the value of A received from Jens K. Andersen we have: For n = 23, a(23) = 313 990 383 602 932 052 632 553 770 22009. - Luis Rodriguez-Torres (ludovicusmagister(AT)yahoo.com), Feb 10 2009

Jens Kruse Andersen. Personal communication (Feb 2009). [Luis Rodriguez-Torres (ludovicusmagister(AT)yahoo.com), Feb 10 2009]
O. Ore, Theory of Numbers and Its History. McGraw Hill, 1948.

C. K. Caldwell, Mills' Theorem - a generalization
Carlos Rivera, Puzzle 85

Cf. A051254, A108739, A051021, A060449, A191357.

a(n) = floor(A^(C^n)); A = 2.084551112... ; C = 1.221. - Luis Rodriguez-Torres (ludovicusmagister(AT)yahoo.com), Feb 10 2009

A191357 Floor(A^(C^n)), where A = 32.76 and C = 1.33.

Original entry on oeis.org

103, 479, 3673, 55147, 2024063, 243937297, 142915724779, 685893080269745, 53978528420922581864, 175329092084368391071206608, 80227969100540338877503013472650510, 26469961649988241699181245714190498215773679043

Offset: 1

Arkadiusz Wesolowski, May 31 2011

nonn

First seven terms are primes.

			a(2) = 479 because 32.76^(1.33^2) = 479.1724192479....

Chris Caldwell, A proof of a generalization of Mills' Theorem
G. L. Honaker, Jr. and Chris Caldwell, Prime Curios! 142915724779
Carlos Rivera, Puzzle 85
Eric Weisstein's World of Mathematics, Floor Function

Cf. A051254, A108739, A051021, A060449, A060699.

default(realprecision, 100); for(n=1, 12, print1(floor(32.76^(1.33^n)), ", ")); \\ Arkadiusz Wesolowski, Jul 18 2011

a(n) = floor(32.76^(1.33^n)).

A224845 Integer lengths of the Mills primes A051254.

Original entry on oeis.org

1, 2, 4, 10, 29, 85, 254, 762, 2285, 6854, 20562, 61684, 185052, 555154, 1665461

Offset: 1

Eric W. Weisstein, Jul 22 2013

Because of the precision of the known Mills' primes and PRPs, it is easy to safely assign decimal lengths of the Mills primes for yet undefined terms (at least another 20-30 terms; they are infinitesimally little offset from successive cubed values). Adding only two terms because these are currently known precisely. - Serge Batalov, Apr 30 2024

			The first few Mills primes are 2, 11, 1361, 2521008887, ... which have integer lengths (= number of decimal digits) of 1, 2, 4, 10, ....

Eric Weisstein's World of Mathematics, Mills' Prime

Cf. A051254 (Mills primes).

Cf. A108739 (b_n associated with Mills primes).

a(14)-a(15) from Serge Batalov, Apr 30 2024

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

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A051254 Mills primes.

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A060449 Generalized Mills numbers: a(n) = floor(c^(b^n)) where c=4.4, b=1.179.

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A060699 a(n) = floor(A^(C^n)), where A = 2.084551112207285611..., C = 1.221.

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A191357 Floor(A^(C^n)), where A = 32.76 and C = 1.33.

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A224845 Integer lengths of the Mills primes A051254.

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

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