A051021 -id:A051021 - 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

A108739 Mills' constant A generates a sequence of primes via b(n)= floor(A^3^n). This sequence is a(n) = b(n+1)-b(n)^3.

Original entry on oeis.org

3, 30, 6, 80, 12, 450, 894, 3636, 70756, 97220, 66768, 300840, 1623568, 8436308

Offset: 1

Chris K. Caldwell, Jun 22 2005

This allows larger terms of A051254 (which triple in digits each entry) to be given. Like A051254, currently requires Riemann Hypothesis to show sequence continues.
Currently a(11)=66768 generates only a probable prime number. - Arkadiusz Wesolowski, May 28 2011
Likewise a(12) and a(13) generate only a probable prime numbers, as well as being conditional on a(11) and a(12) being proved primes. Minimality of a(12)-a(13) is exhaustively tested. - Serge Batalov, Aug 06 2013
a(14) = 8436308 is found by Ryan Propper and Serge Batalov, Apr 29 2024, but a few remaining gaps below this value were being double-checked. The double-check is now complete (see GitHub link). - Ryan Propper and Serge Batalov, May 24 2024.

			The Mills' primes (given in A051254) are 2, 2^3+3 = 11, (2^3+3)^3+30 = 11^3+30 = 1361, ((2^3+3)^3+30)^3+6 = 1361^3+6 = 2521008887, etc. The terms added at each step yield this sequence. They are the least positive integers which added to the cube of the preceding prime yield again a prime, cf. formula. - _M. F. Hasler_, Jul 22 2013

T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 8.

Chris K. Caldwell, Mills' Theorem - a generalization.
Chris K. Caldwell, The List of Largest Known Primes, The 11th Mills' prime
Chris K. Caldwell and Yuanyou Cheng, Determining Mills' Constant and a Note on Honaker's Problem, Journal of Integer Sequences, Vol. 8 (2005), Article 05.4.1.
Sergey Batalov, Mills' b(14) search report
Henri & Renaud Lifchitz, PRP Records and PRP Records, search:Mills
W. H. Mills, A prime-representing function, Bull. Amer. Math. Soc., Vol. 53 (1947), p. 604.
Eric Weisstein's World of Mathematics, Mills' Constant
Eric Weisstein's World of Mathematics, Mills' Prime
E. M. Wright, A class of representing functions, J. London Math. Soc., Vol. 29 (1954) pp. 63-71.

Cf. A051254, A051021.

B[1] = 2; B[n_] := B[n] = NextPrime[B[n - 1]^3]; Table[B[n + 1] - B[n]^3, {n, 7}] (* Robert Price, Jun 09 2019 *)

p=2; until(, np=nextprime(p^3); print1(np-p^3, ", "); p=np) \\ Jeppe Stig Nielsen, Apr 22 2020

b(1) = 2; b(n+1) = nextprime(b(n)^3); a(n) = b(n+1)-b(n)^3;

a(9)-a(11) from Caldwell and Cheng, Aug 29 2005
Corrected by T. D. Noe, Sep 24 2008
a(12) (which generates a PRP) from Serge Batalov, Jul 19 2013
a(13) (which generates a PRP) from Serge Batalov, Aug 06 2013
a(14) (which generates a PRP) from Ryan Propper and Serge Batalov, May 24 2024

A338613 Numbers given by a(n) = 1 + floor(c^(n^1.5)) where c=2.2679962677... is the constant defined at A338837.

Original entry on oeis.org

2, 3, 11, 71, 701, 9467, 168599, 3860009, 111498091, 4002608003, 176359202639, 9437436701437, 607818993573569, 46744099128452807, 4262700354254812091, 458091929703695291747, 57691186909930154615407, 8471601990692484416847631, 1443868262009075144775972529

Offset: 0

Bernard Montaron, Nov 03 2020

nonn

Assuming Cramer's conjecture on largest prime gaps, it can be proved that there exists at least one constant 'c' such that all a(n) are primes for n as large as required. The constant giving the smallest growth rate is c=2.2679962677067242473285532807253717745270422544...
This exponential sequence of prime numbers grows very slowly compared to Mills' sequence for which each new term has 3 times more digits than the previous one. More than 60 terms (all prime numbers) can be easily calculated for the sequence described here which is quite remarkable for an exponential sequence.
Algorithm to compute the smallest constant 'c' and the associated prime number sequence a(n).
  n=0, a(0)=2, c=2, d=1.5
  n=n+1
  b=1+floor(c^(n^d))
  p=smpr(b)     smallest prime >= b
  If p=b then a(n)=p, go to 1.
  c=(p-1)^(1/n^d)
  a(n)=p
  k=1
  b=1+floor(c^(k^d))
  If b<>a(k) then p=smpr(b), n=k, go to 5.
If k11. go to 1.
I propose the following generalization:  find the function f(n) with f(0)=0 and f(x)>x for x>=2 such that there exists a suitable positive constant c(f) giving the increasing prime sequence a(n)=1+floor(c^f(n)) with the smallest possible growth rate. Since a(0)=2, c(f)>=2.

François Marques, Table of n, a(n) for n = 0..199
Bernard Montaron, Exponential prime sequences, arXiv:2011.14653 [math.NT], 2020.

Cf. A338837, A338850, A051021, A051254.

c(n=40, prec=100)={
  my(curprec=default(realprecision));
  default(realprecision, max(prec, curprec));
  my(a=List([2]), d=1.5, c=2.0, b, p, ok, smpr(b)=my(p=b); while(!isprime(p), p=nextprime(p+1)); return(p); );
  for(j=1, n-1,
    b=1+floor(c^(j^d));
    until(ok,
      ok=1;
      p=smpr(b);
      listput(a,p,j+1);
      if(p!=b,
         c=(p-1)^(j^(-d));
         for(k=1,j-2,
             b=1+floor(c^(k^d));
             if(b!=a[k+1],
                ok=0;
                j=k;
                break;
               );
            );
        );
    );
  );
  default(realprecision, curprec);
  return(a);
} \\ François Marques, Nov 12 2020

a(n) = 1 + floor(c^(n^1.5)) where c=2.2679962677...

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

A117739 Decimal expansion of the largest C_0 = 1.2209864... such that for C < C_0 and A < 2 the sequence a(n) = floor[A^(C^n)] can't contain only prime terms.

Original entry on oeis.org

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

Offset: 1

Martin Raab, May 04 2006

It is not proved that for C > C_0 the mentioned infinite sequence of primes actually exists. However, heuristics show that A243358 could be infinite (the decimal expansion of corresponding A value is A243370).

Andrey V. Kulsha, Table of n, a(n) for n = 1..50000
Chris K. Caldwell, A proof of a generalization of Mills' Theorem

Cf. A243358 (primes), A243370 (value of A), A051021 (Mills' constant)

C_0 can be estimated as (logP/log84)^(1/k), where P is k+10th term of A243358.

Terms after a(18) from Andrey V. Kulsha, Jun 03 2014

A112597 Decimal expansion of x, where x is the smallest number for which floor(x^(2^y)) is prime for every y > 0 (assuming the truth of Legendre's conjecture).

Original entry on oeis.org

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

Offset: 1

Martin Raab, Dec 21 2005

			1.524699960538...
Repeated squaring gives the primes 2, 5, 29, 853, 727613, 529420677791, 280286254072681840639693, ... (A059784).

Cf. A059784, A051021, decimal expansion of Mills' constant where floor(x^(3^y)) is prime for every y > 0.

Name clarified by Thomas Scheuerle, Mar 28 2025

A079614 Decimal expansion of Bertrand's constant.

Original entry on oeis.org

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

Offset: 1

Benoit Cloitre, Jan 29 2003

From Bertrand's postulate (i.e., there is always a prime p in the range n < p < 2n) one can show there is a constant b such that floor(2^b), floor(2^2^b), ..., floor(2^2^2...^b), ... are all primes.

This result is due to Wright (1951), so Bertrand's constant might be better called Wright's constant, by analogy with Mills's constant A051021. - Jonathan Sondow, Aug 02 2013

			2^(2^(2^1.251647597790463017594432053623)) is approximately 37.0000000000944728917062132870071 and A051501(3)=37.

S. Finch, Mathematical Constants, Cambridge Univ. Press, 2003; see section 2.13 Mills's constant.

C. K. Caldwell, Prime Curios! 137438953481.
Pierre Dusart, Estimates of some functions over primes without R. H., arXiv:1002.0442 [math.NT], 2010.
J. Sondow, E. Weisstein, Bertrand's Postulate.
E. M. Wright, A prime-representing function, Amer. Math. Monthly, 58 (1951), 616-618.

Cf. A051021, A051501, A060715.

1.251647597790463017594432053623346969...

More digits (from the Prime Curios page) added by Frank Ellermann, Sep 19 2011
a(16)-a(37) from Charles R Greathouse IV, Sep 20 2011
Definition clarified by Jonathan Sondow, Aug 02 2013

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)).

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]]

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A051254 Mills primes.

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A108739 Mills' constant A generates a sequence of primes via b(n)= floor(A^3^n). This sequence is a(n) = b(n+1)-b(n)^3.

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A338613 Numbers given by a(n) = 1 + floor(c^(n^1.5)) where c=2.2679962677... is the constant defined at A338837.

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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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A117739 Decimal expansion of the largest C_0 = 1.2209864... such that for C < C_0 and A < 2 the sequence a(n) = floor[A^(C^n)] can't contain only prime terms.

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A112597 Decimal expansion of x, where x is the smallest number for which floor(x^(2^y)) is prime for every y > 0 (assuming the truth of Legendre's conjecture).

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A079614 Decimal expansion of Bertrand's constant.

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