A046029 -id:A046029 - cp's OEIS frontend

A200906 Numbers n such that cyclotomic polynomial value Phi(5,n!) is prime.

0, 1, 2, 4, 5, 21, 44, 64, 244, 268, 2415

Offset: 1

Serge Batalov, Nov 23 2011

nonn

2415 corresponds to a probable prime. - Serge Batalov, Nov 24 2011

a(12) > 15000. - Robert Price, Jun 20 2015

			5 is in the sequence because Phi(5,5!) = ((5!)^5-1)/(5!-1)= 209102521 is prime.

Cf. A002981, A002982, A046029, A046029.

Do[If[PrimeQ[Cyclotomic[5, n!]], Print[n]], {n, 0, 600}]

for(n=0,600,x=n!;if(isprime(eval(polcyclo(5))),print(n)))

A051856 Numbers k such that (k!)^2 + k! + 1 is prime.

Original entry on oeis.org

0, 1, 2, 3, 4, 6, 76, 2837, 6001, 7076

Offset: 1

Andrew Walker (ajw01(AT)uow.edu.au), Dec 13 1999

			6 is in the sequence because (6!)^2+6!+1=519121 is prime.

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

C. K. Caldwell, The Prime Pages
C. Nash, Prime Form [broken link]
M. Oakes, Re: Gaussian primorial and factorial primes, Primeform, Dec 21 2010 [broken link]
Mike Oakes, Andrew Walker, David Broadhurst, Gaussian primorial and factorial primes, digest of 7 messages in primeform Yahoo group, Dec 20 - Dec 21, 2010.

Cf. A002981, A002982, A046029.

Do[If[PrimeQ[n!^2+n!+1], Print[n]], {n, 600}] (* Farideh Firoozbakht, Jul 12 2003 *)

Edited by R. J. Mathar, Aug 08 2008

a(8)-a(10) from Serge Batalov, Nov 24 2011

A282706 Smallest prime factor of A020549(n) = (n!)^2 + 1.

Original entry on oeis.org

2, 2, 5, 37, 577, 14401, 13, 101, 17, 131681894401, 13168189440001, 1593350922240001, 101, 38775788043632640001, 29, 1344169, 149, 9049, 37, 710341, 41, 61, 337, 509, 384956219213331276939737002152967117209600000001, 941

Offset: 0

N. J. A. Sloane, Feb 26 2017

nonn

By construction, for n >= 2, a(n) == 1 (mod 4) and a(n) > n.
From Robert Israel, Mar 08 2017: (Start)
a(n) = A020549(n) for n in A046029.
a(n) <= 2*n+1 if n is in A104636.
The first member of A104636 for which a(n) < 2*n+1 is 48.
a(a(n)-n-1) = a(n). (End)

T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 147.

Hugo Pfoertner, Table of n, a(n) for n = 0..74
Apoloniusz Tyszka, On ZFC-formulae phi(x) for which we know a non-negative integer n such that max({x, element of N, phi(x)}) <= n if the set {x, element of N, phi(x)} is finite, 2019.
Apoloniusz Tyszka, Sławomir Kurpaska, Open problems that concern computable sets X, subset of N, and cannot be formally stated as they refer to current knowledge about X, (2020).
Andrew Walker, Table of factors of (n!)^2+1.

Cf. A020549, A046029, A083340, A083341, A104636, A282705.

[2] cat [Min(PrimeFactors(Factorial(n)^2 + 1)):n in[1..25]]; // Vincenzo Librandi, Feb 28 2017

f:= proc(n) local a;
  a:= min(map(proc(t) if t[1]::integer then t[1] fi end proc, ifactors((n!)^2+1,easy)[2]));
if a = infinity then
   a:= traperror(timelimit(60, min(map(t -> t[1], ifactors((n!)^2+1)[2]))));
fi;
  a
end proc:
map(f, [$0..36]); # Robert Israel, Mar 08 2017

Join[{2}, Array[FactorInteger[(#!)^2 + 1][[1, 1]]&, {25}]] (* Vincenzo Librandi, Feb 28 2017 *)

More terms from Vincenzo Librandi, Feb 28 2017

A051739 Primes of the form (n!)^2 + 1.

Original entry on oeis.org

2, 5, 37, 577, 14401, 131681894401, 13168189440001, 1593350922240001, 38775788043632640001, 384956219213331276939737002152967117209600000001

Offset: 1

G. L. Honaker, Jr., Dec 06 1999

nonn

			37 is a term because it is prime and is (3!)^2 + 1.

Vincenzo Librandi, Table of n, a(n) for n = 1..12

Cf. A046029. Primes in A020549.

[a: n in [1..50] | IsPrime(a) where a is  Factorial(n)^2+1]; // Vincenzo Librandi, Dec 08 2011

lst={};Do[s=n!^2;If[PrimeQ[p=s+1], AppendTo[lst, p]], {n, 5!}];lst (* Vladimir Joseph Stephan Orlovsky, Sep 27 2008 *)
Select[Table[(n!)^2+1,{n,1,5000}],PrimeQ] (* Vincenzo Librandi, Dec 08 2011 *)

More terms from James Sellers, Dec 08 1999

A051855 Numbers n such that (n!)^4+1 is prime.

Original entry on oeis.org

0, 1, 2, 3, 4, 13, 112, 328, 11123

Offset: 1

Andrew Walker (ajw01(AT)uow.edu.au), Dec 13 1999

nonn

C. K. Caldwell, The Prime Pages
C. Nash, Prime Form [?Broken link]

Cf. A002981, A002982, A046029.

[n: n in [1..300] | IsPrime(Factorial(n)^4+1)]; // Vincenzo Librandi, Aug 15 2013

Select[Range[0, 350], PrimeQ[(#!)^4 + 1]&] (* Vincenzo Librandi, Aug 15 2013 *)

isok(n) = isprime(n!^4 + 1); \\ Michel Marcus, Aug 15 2013

a(9) from Robert Price, Jul 24 2014

Prepended a(1)=0, Robert Price, Sep 01 2014

A083340 Numbers n such that A020549(n)=(n!)^2+1 is a semiprime.

Original entry on oeis.org

6, 7, 8, 12, 15, 16, 17, 18, 19, 28, 29, 41, 45, 53, 55, 61, 73

Offset: 1

Hugo Pfoertner, Apr 24 2003

The smaller of the two prime factors is given in A083341. The next candidates for a continuation are 55 and 61. (55!)^2 + 1 is composite with 147 decimal digits and unknown factorization.
(55!)^2 + 1 has been factored using ECM into P52*P96 with P52 = A083341(15). (61!)^2 + 1 is composite with 168 decimal digits. - Hugo Pfoertner, Jul 13 2019
Using CADO-NFS, (61!)^2 + 1 has been factored into P58*P110 with P58 = A282706(61) in 17 days wall clock time using 56 million CPU seconds. a(18) >= 75. - Hugo Pfoertner, Aug 04 2019

			a(1)=6 because (6!)^2+1=518401=13*39877 is a semiprime.

The CADO-NFS Development Team, Cado-NFS, An Implementation of the Number Field Sieve Algorithm, Release 2.3.0, 2017
Andrew Walker, Factors of (n!)^2+1.
factordb.com, Status of 75!^2+1.

Cf. A020549, A001358, A083341, A046029, A282706.

Select[Range[60],PrimeOmega[(#!)^2+1]==2&] (* Harvey P. Dale, Dec 12 2018 *)

a(15) from Hugo Pfoertner, Jul 13 2019

a(16), a(17) from Hugo Pfoertner, Aug 04 2019

A359180 Numbers k such that k!^2 / 2 + 1 is prime.

Original entry on oeis.org

2, 3, 6, 18, 19, 82, 1298, 3139, 3687, 4637

Offset: 1

Arsen Vardanyan, Dec 18 2022

			3!^2 / 2 + 1 = 6^2/2 + 1 = 19, a prime number, so 3 is a term.

Cf. A002981, A046029, A358805, A358878.

isok(k) = (k>1) && isprime(k!^2 / 2 + 1); \\ Michel Marcus, Jan 15 2023

a(7) from Michael S. Branicky, Dec 18 2022

a(8)-a(10) from Michael S. Branicky, Apr 10 2023

A053608 Numbers x = LCM(1,2,...,k) such that x^2 + 1 is prime.

Original entry on oeis.org

1, 2, 6, 420, 360360, 718766754945489455304472257065075294400

Offset: 1

Labos Elemer, Feb 09 2000

The next term has k > 10^4, if it exists. - Amiram Eldar, Aug 23 2024

Cf. A003418, A049536, A046029, A051418, A051451, A051739, A053609.

Select[FoldList[LCM, 1, Select[Range[100], PrimePowerQ]], PrimeQ[#^2 + 1] &] (* Amiram Eldar, Aug 23 2024 *)

A053609 Primes of form x^2+1 where x = LCM(1,2,...,k) for some k.

Original entry on oeis.org

2, 5, 37, 176401, 129859329601, 516625648014869290354797521879383114125823989794742396526049715541246671360001

Offset: 1

Labos Elemer, Feb 09 2000

The next term has k > 10^4, if it exists. - Amiram Eldar, Aug 23 2024

Cf. A049536, A046029, A051739, A051418, A051451, A053608.

Select[FoldList[LCM, 1, Select[Range[100], PrimePowerQ]]^2 + 1, PrimeQ] (* Amiram Eldar, Aug 23 2024 *)

a(n) = A053608(A053608(n)) = A053608(n)^2 + 1. - Amiram Eldar, Aug 23 2024

A064769 Numbers n such that (n!)^2 + prime(n) is prime.

Original entry on oeis.org

1, 2, 3, 5, 9, 17, 65, 222, 1720, 2975, 3494, 10489, 17948

Offset: 1

Jason Earls, Oct 18 2001

nonn

Term <= 1720 certified prime with Primo.
Next term, if it exists, is greater than 2700. - Ryan Propper, Nov 05 2005
a(10)-a(13) are probable primes.
a(14) > 20000. - Robert Price, Aug 29 2014

			(5!)^2 + prime(5) = 120^2 + 11 = 14400 + 11 = 14411 is prime, so 5 is a term.

Cf. A046029, A072599 (n such that (n!)^2 - prime(n) is prime).

for(n=1,300, if(isprime((n!)^2+prime(n)),print1(n, ", ")))

1720 from Ryan Propper, Nov 05 2005

a(10)-a(13) from Robert Price, Aug 29 2014

cp's OEIS Frontend

A200906 Numbers n such that cyclotomic polynomial value Phi(5,n!) is prime.

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A051856 Numbers k such that (k!)^2 + k! + 1 is prime.

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A282706 Smallest prime factor of A020549(n) = (n!)^2 + 1.

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A051739 Primes of the form (n!)^2 + 1.

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A051855 Numbers n such that (n!)^4+1 is prime.

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A083340 Numbers n such that A020549(n)=(n!)^2+1 is a semiprime.

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A359180 Numbers k such that k!^2 / 2 + 1 is prime.

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A053608 Numbers x = LCM(1,2,...,k) such that x^2 + 1 is prime.

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