A076656 -id:A076656 - 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

A229610 Array: each row starts with the least prime not in a previous row, and each prime p in a row is followed by the least prime > 3*p.

Original entry on oeis.org

2, 7, 3, 23, 11, 5, 71, 37, 17, 13, 223, 113, 53, 41, 19, 673, 347, 163, 127, 59, 29, 2027, 1049, 491, 383, 179, 89, 31, 6089, 3163, 1481, 1151, 541, 269, 97, 43, 18269, 9491, 4447, 3457, 1627, 809, 293, 131, 47, 54829, 28477, 13367, 10391, 4889, 2437, 881

Offset: 1

Clark Kimberling, Sep 26 2013

Conjectures: (row 1) = A076656, (column 1) = A164958, and for each row r(k), the limit of r(k)/3^k exists. For rows 1 to 4, the respective limits are 0.928655..., 1.447047..., 2.038260..., 4.753271... .

			Northwest corner:
   2,  7,  23,  71,  223,  673, ...
   3, 11,  37, 113,  347, 1049, ...
   5, 17,  53, 163,  491, 1481, ...
  13, 41, 127, 383, 1151, 3457, ...
  19, 59, 179, 541, 1627, 4889, ...
  29, 89, 269, 809, 2437, 7331, ...

Cf. A076656, A164958, A229607, A229608, A229609.

seqL = 14; arr2[1] = {2}; Do[AppendTo[arr2[1], NextPrime[3*Last[arr2[1]]]], {seqL}]; Do[tmp = Union[Flatten[Map[arr2, Range[z]]]]; arr2[z] = {Prime[NestWhile[# + 1 &, 1, PrimePi[tmp[[#]]] - # == 0 &]]}; Do[AppendTo[arr2[z], NextPrime[3*Last[arr2[z]]]], {seqL}], {z, 2, 12}]; m = Map[arr2, Range[12]]; m // TableForm
t = Table[m[[n - k + 1]][[k]], {n, 12}, {k, n, 1, -1}] // Flatten (* Peter J. C. Moses, Sep 26 2013 *)

Incorrect comment deleted by Peter Munn, Aug 15 2017

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

A118476 a(0) = 1; a(n) is least k with n prime factors and k > n*a(n-1).

Original entry on oeis.org

1, 2, 6, 20, 81, 408, 2480, 17376, 139040, 1251450, 12514816, 137663064, 1651956992, 21475443200, 300656206080, 4509843098112, 72157489576704, 1226677322842112, 22080191811166208, 419523644412176256, 8390472888243683328, 176199930653117513728

Offset: 0

Jonathan Vos Post, May 04 2006

This is a super-polynomial function, as for positive n, a(n) > n!.

Prime factors counted with multiplicity. - Harvey P. Dale, Aug 25 2019

			a(1) = 2 because 2 is the smallest prime (integer with 1 prime factor) greater than 1 * 1 = 1.
a(2) = 6 because 6 = 2 * 3 is the smallest semiprime (integer with 2 prime factors) greater than 2 * 2 = 4.
a(3) = 20 because 20 = 2^2*5 is the smallest 3-almost prime (integer with 3 prime factors) greater than 3 * 6 = 18.

Eric Weisstein's World of Mathematics, Almost Prime.

Cf. A000040, A001358, A055496, A076656.

A118476 := proc(n) option remember; local k; if n = 0 then 1; else for k from n*procname(n-1)+1 do if numtheory[bigomega](k) = n then return k; end if; end do: end if; end proc:
seq(A118476(n),n=0..14) ; # R. J. Mathar, Dec 22 2010

lkpf[{n_,a_}]:=Module[{k=a(n+1)+1},While[PrimeOmega[k]!=n+1,k++];{n+1,k}]; NestList[lkpf,{0,1},21][[All,2]] (* Harvey P. Dale, Aug 25 2019 *)

a(0) = 1; a(n) least n-almost prime > n*a(n-1).

Terms corrected from a(4) on by R. J. Mathar, Dec 22 2010

a(15)-a(21) from Donovan Johnson, Jan 06 2011

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.

A117880 a(1) = 4; a(n) is smallest semiprime > 2*a(n-1).

Original entry on oeis.org

4, 9, 21, 46, 93, 187, 377, 755, 1513, 3027, 6059, 12127, 24257, 48529, 97059, 194127, 388257, 776515, 1553033, 3106083, 6212177, 12424355, 24848723, 49697447, 99394909, 198789819, 397579639, 795159283, 1590318573, 3180637153

Offset: 1

Jonathan Vos Post, May 04 2006

a(1)=4, a(n)=2*a(n-1)+k, where k is least positive integer chosen so that a(n) is the product of two primes. Corresponding k's are 1, 3, 4, 1, 1, 3, 1, 3, 1, 5, 9, 3, 15, 1, 9, 3, 1, 3, 17, 11, 1, 13, 1, 15, 1, 1, 5, 7, 7, 11, 5, 5, 15, 1, 3, 9, 9, 5, 7, 8, ... - Zak Seidov, Dec 24 2007

			a(1)=4, then
k=1, a(2)=2*4+1=9,
k=3, a(3)=2*9+3=21,
k=4, a(4)=2*21+4=46,
k=1, a(5)=2*46+1=93,
k=1, a(6)=2*93+1=187.

Semiprime analog of A055496.

Cf. A001358, A076656.

a=0;Do[Do[b=2a+n;If[2==Plus@@FactorInteger[b][[All,2]],Print[b];Break[]],{n,1000}];a=b,{40}] (* Zak Seidov, Dec 24 2007 *)
ssp[n_]:=Module[{k=2n+1},While[PrimeOmega[k]!=2,k++];k]; NestList[ssp,4,30] (* Harvey P. Dale, Apr 14 2022 *)

Edited by N. J. A. Sloane, Jul 01 2008 at the suggestion of R. J. Mathar

A117916 a(1) = 4; a(n) is smallest semiprime > 3*a(n-1).

Original entry on oeis.org

4, 14, 46, 141, 427, 1282, 3849, 11553, 34663, 104001, 312005, 936017, 2808053, 8424161, 25272487, 75817463, 227452391, 682357183, 2047071551, 6141214658, 18423643982, 55270931959, 165812795887, 497438387665, 1492315162999

Offset: 1

Jonathan Vos Post, May 04 2006

Semiprime analog of A076656 a(1) = 2; a(n) is smallest prime > 3*a(n-1). a(n)-a(n-1) is often 1, never more than 16 through n = 28, then jumps to 32 for n = 29; and I wonder what its value or mean value is asymptotically as a function of n.

Cf. A001358, A055496, A076656.

nxt[n_]:=Module[{c=3n,k=1},While[PrimeOmega[c+k]!=2,k++];c+k]; NestList[ nxt,1,30] (* Harvey P. Dale, May 31 2012 *)

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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A229610 Array: each row starts with the least prime not in a previous row, and each prime p in a row is followed by the least prime > 3*p.

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

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A118476 a(0) = 1; a(n) is least k with n prime factors and k > n*a(n-1).

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

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A117880 a(1) = 4; a(n) is smallest semiprime > 2*a(n-1).

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