A051254 -id:A051254 - cp's OEIS frontend

A051021 Decimal expansion of Mills's constant, assuming the Riemann Hypothesis is true.

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

Offset: 1

Eric W. Weisstein

Not known to be rational or irrational. See Saito (2024) for a new result. - Charles R Greathouse IV, Jul 18 2013, Hugo Pfoertner, May 01 2024

			1.3063778838630806904686144926026057129167845851567136443680537599664340537668...

T. 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..10000 (first 641 terms from Tin Apato)
C. K. Caldwell, Mills's Constant [Gives 6000 terms assuming the Riemann Hypothesis.]
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.
Christian Elsholtz, Unconditional Prime-representing Functions, Following Mills, arXiv:2004.01285 [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.
Aminu Alhaji Ibrahim and Sa’idu Isah Abubaka, Aunu Integer Sequence as Non-Associative Structure and Their Graph Theoretic Properties, Advances in Pure Mathematics, 2016, 6, 409-419.
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.
Bernard Montaron, Exponential prime sequences, arXiv:2011.14653 [math.NT], 2020.
Robert P. Munafo, Notable Properties of Specific Numbers.
Simon Plouffe, The calculation of p(n) and pi(n), arXiv:2002.12137 [math.NT], 2020.
Kota Saito, Mills' constant is irrational, arXiv:2404.19461 [math.NT], 2024.
László Tóth, A Variation on Mills-Like Prime-Representing Functions, arXiv:1801.08014 [math.NT], 2018.
Eric Weisstein's World of Mathematics, Mills' Constant.
Eric Weisstein's World of Mathematics, Prime Formulas.

Cf. A051254.

RealDigits[ Nest[ NextPrime[#^3] &, 2, 7]^(1/3^8), 10, 111][[1]] (* Robert G. Wilson v, Nov 14 2012 *)

A051021_upto(N=99)=localprec(N+9);digits(10^N*sqrtn(A051254(N=logint(N,3)+2),3^N)\1) \\ M. F. Hasler, Sep 11 2024

More terms from Robert G. Wilson v, Sep 08 2000

More terms from Tin Apato (tinapto(AT)yahoo.es), Dec 12 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...

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

Original entry on oeis.org

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

A063636 a(n) = floor((1287/545)^n).

Original entry on oeis.org

2, 5, 13, 31, 73, 173, 409, 967, 2283, 5392, 12735, 30073, 71017, 167706, 396032, 935217, 2208486, 5215270, 12315692, 29083113, 68678837, 162182870, 382989640, 904417737, 2135753445, 5043513182, 11910094433, 28125305569, 66417005997

Offset: 1

Jud McCranie, Aug 10 2001

nonn

The first eight terms are primes. Does there exist a number theta such that the floor of theta^n is always prime?

			(1287/545)^3 = 13.16879..., so a(3)=13.

Richard Crandall and Carl Pomerance, Prime Numbers - a Computational Perspective, Springer, 2001, page 69, exercise 1.75.

Harry J. Smith, Table of n, a(n) for n = 1..300

Cf. A000040, A051254, A060449, A060699.

{ for (n=1, 300, write("b063636.txt", n, " ", 1287^n \ 545^n); ) } \\ Harry J. Smith, Aug 26 2009

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

A118910 a(1) = 2; a(n) is greatest prime < a(n-1)^3.

Original entry on oeis.org

2, 7, 337, 38272739, 56062005704198360319209, 176199995814327287356671209104585864397055039072110696028654438846269

Offset: 1

Jonathan Vos Post, May 05 2006

Exponent 3 analog of A059785.

Obverse of this is A051254.

			a(5) = 62343227157465615355481 = a(4)^3 - 32 = 39651817^3 - 32 and there is no k < 32 such that 39651817^3 - k is prime.

Cf. A001358, A055496, A076656, A006992, A005384, A005385, A118908-A118913.

a=2; Join[{2}, Table[a=a^3; While[ !PrimeQ[a], a=a-1]; a, {5}]] (* T. D. Noe, Nov 15 2006 *)

Corrected by T. D. Noe, Nov 15 2006

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)).

Showing 1-10 of 16 results. Next

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

cp's OEIS Frontend

A224845 Integer lengths of the Mills primes A051254.

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A051021 Decimal expansion of Mills's constant, assuming the Riemann Hypothesis is true.

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

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A063636 a(n) = floor((1287/545)^n).

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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.