A059785 -id:A059785 - cp's OEIS frontend

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

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

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

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]]

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.

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Examples

Crossrefs

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

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Extensions

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

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Python

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.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

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

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula