A118913 -id:A118913 - 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

A118908 a(1) = 4; a(n) is greatest semiprime < a(n-1)^2.

Original entry on oeis.org

4, 15, 221, 48839, 2385247913, 5689407606470855563, 32369358912568429679140929317208046943, 1047775396410673232345014594095988998399127191704501568910205139392491645247

Offset: 1

Jonathan Vos Post, May 05 2006

Semiprime analog of A059785 a(n+1)=prevprime(a(n)^2), with a(1) = 2. With that, of course, there's always a prime between n and 2n, so a(n) < 2^n. See also A055496 a(1) = 2; a(n) is smallest prime > 2*a(n-1). The obverse of this is A118909 a(1) = 4; a(n) is least semiprime > a(n-1)^2.

a(9), which is too large to be included, is equal to a(8)^2-3. - Giovanni Resta, Jun 16 2016

			a(6) = 32369358912568429679140929317208046943 = 1816568472934912211 * 17818958874845686213 = 5689407606470855563^2 - 26 < a(5)^2.

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

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

A087147 Numbers k such that k! + (k+1)! + 1 is prime.

Original entry on oeis.org

0, 3, 7, 9, 67, 291, 1343, 6984, 12861

Offset: 1

Farideh Firoozbakht, Aug 19 2003

291 is in the sequence and also is in the sequence A087146, thus (291!+292!-1,291!+292!+1) is a twin pair of primes. Any additional terms are greater than 1800 with the next prime having more than 5086 digits.
Next term is greater than 4200. - Gabriel Cunningham (gcasey(AT)mit.edu), Sep 09 2003
a(10) > 25000. - Robert Price, Aug 26 2015
k+1 is not prime because (p-1)! + p! + 1 == 0 mod p and (p-1)! + p! + 1 > p where p is prime. - Seiichi Manyama, Mar 22 2018

			3 is in the sequence because 3!+4!+1=31 is prime.

H. Dubner, Factorial and primorial primes, J. Rec. Math., 19 (No.3, 1987)

Eric Weisstein's World of Mathematics, Wilson's Theorem

Cf. A087146, A051856.

Primes in A118913. [From Dmitry Kamenetsky, Oct 21 2008]

v={}; Do[If[PrimeQ[n!+(n+1)!+1], v=Append[v, n]; Print[v]], {n, 1800}]; v
Select[Range[0,25000],PrimeQ[#!+(#+1)!+1]&] (* Robert Price, Aug 26 2015 *)

isok(k) = ispseudoprime(k!+(k+1)!+1); \\ Altug Alkan, Mar 22 2018

a(8)-a(9) from Robert Price, Aug 26 2015

A118909 a(1) = 4; a(n) is least semiprime > a(n-1)^2.

Original entry on oeis.org

4, 21, 445, 198026, 39214296677, 1537761063871773242347, 2364709089560047865452947255794201194068433, 5591849078247910476736920566826713466552016538943524658263883555662554776622687075541

Offset: 1

Jonathan Vos Post, May 05 2006

Semiprime analog of A055496 a(1) = 2; a(n) is smallest prime > 2*a(n-1). See also A059785 a(n+1)=prevprime(a(n)^2), with a(1) = 2. With that, of course, there's always a prime between n and 2n, so a(n) < 2^n. The obverse of this is A118908 a(1) = 4; a(n) is greatest semiprime < a(n-1)^2.

			a(8) = a(7)^2 + 52 and there is no smaller k such that a(7)^2 + k is semiprime.

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

nxt[n_]:=Module[{sp=n^2+1},While[PrimeOmega[sp]!=2,sp++];sp]; NestList[nxt,4,7] (* Harvey P. Dale, Oct 22 2012 *)

from itertools import accumulate
from sympy.ntheory.factor_ import primeomega
def nextsemiprime(n):
  while primeomega(n + 1) != 2: n += 1
  return n + 1
def f(anm1, _): return nextsemiprime(anm1**2)
print(list(accumulate([4]*6, f))) # Michael S. Branicky, Apr 21 2021

A118912 a(1) = 2; a(n) is greatest prime < a(n-1)^4.

Original entry on oeis.org

2, 13, 28559, 665230244078823349, 195833931687186822327230545227550596864953022841534058316595001440791433

Offset: 1

Jonathan Vos Post, May 05 2006

Exponent-4 analog of A059785 a(n+1)=prevprime(a(n)^2), with exponent 3 being A118910 a(1) = 2; a(n) is greatest prime < a(n-1)^3.

			a(1) = 2, by definition.
a(2) = 13 = 2^4 - 3.
a(3) = 28559 = 13^4 - 2.
a(4) = 665230244078823349 = 28559^4 - 12.
a(5) = 195833931687186822327230545227550596864953022841534058316595001440791433 = 665230244078823349^4 - 168.
a(6) is too large to include.

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

NestList[NextPrime[#^4,-1]&,2,5] (* Harvey P. Dale, Feb 18 2025 *)

a(1) = 2; a(n) is greatest prime < a(n-1)^4.

A125174 Primes of the form k! + (k+1)! - 1.

Original entry on oeis.org

2, 7, 29, 839, 3991679, 93405311999, 1394852659199, 6758061133823999, 315777214062132212662271999999, 9146650338351415815045119999999, 303916116658416027343136804044799999999, 836313165329095177704251551336018791628799999999

Offset: 1

Tomas Xordan, Jan 12 2007

nonn

Next term is too long (167 digits) to include in data. - James C. McMahon, Dec 11 2024

			0!+1!-1 = 1 is not prime, 1!+2!-1 = 2 is prime, 2!+3!-1 = 7 is prime, 3!+4!-1 = 29 is prime, 4!+5!-1 = 143 = 11*13 is not prime.

Cf. A087146 (values of k), A118913 (primes of the form k! + (k+1)! + 1).

Select[Table[n!+(n+1)!-1,{n,110}],PrimeQ] (* James C. McMahon, Dec 11 2024 *)

for(n=0,35,if(isprime(a=n!+(n+1)!-1),print1(a,",")))

Edited and extended by Klaus Brockhaus, Jan 24 2007

a(12) from James C. McMahon, Dec 11 2024

cp's OEIS Frontend

A051254 Mills primes.

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A118908 a(1) = 4; a(n) is greatest semiprime < a(n-1)^2.

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

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A087147 Numbers k such that k! + (k+1)! + 1 is prime.

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A118909 a(1) = 4; a(n) is least semiprime > a(n-1)^2.

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A118912 a(1) = 2; a(n) is greatest prime < a(n-1)^4.

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A125174 Primes of the form k! + (k+1)! - 1.

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