A088332 -id:A088332 - cp's OEIS frontend

A118370 Divisorial primes: Primes p such that p = 1 + Product_{d|n} d for some n (ordered by n).

2, 3, 37, 101, 197, 331777, 677, 8503057, 9834497, 5477, 59969537, 8837, 17957, 21317, 562448657, 916636177, 42437, 3208542737, 3782742017, 5006411537, 7676563457, 98597, 106277, 11574317057, 19565295377, 416806419029812551937, 148997, 34188010001, 38167092497

Offset: 1

Rick L. Shepherd, Apr 25 2006

nonn

See A118369 for the corresponding n. These are primes in the sequence 1 + A007955. (The suggested name "divisorial prime" is obviously analogous to that of factorial primes (A088332) and primorial primes (A014545).).

			The prime 37 is a(3) as there exists a number, A118369(3)=6, such that 37 = 6*3*2*1 + 1, where {1,2,3,6} are all the positive divisors of 6.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A118369, A007955.
Cf. A014545, A088332.
Cf. A258455 (sorted).

Reap[For[n = 1, n <= 500, n++, p = Times @@ Divisors[n]; If[PrimeQ[p+1], Sow[p+1]]]][[2, 1]] (* Jean-François Alcover, Oct 07 2016 *)

for(n=1,2500, s=1; fordiv(n,d,s=s*d); if(isprime(s+1), print1(s+1,", ")))

A300947 Primes of form (2*k)! + k! + 1.

Original entry on oeis.org

3, 727, 20922789928321, 403291461126605641811020801, 523022617466601111760007224221719391608832001

Offset: 1

Seiichi Manyama, Mar 22 2018

nonn

The next term is too large to include.

Cf. A088332, A118812, A242487.

select(isprime,[seq(factorial(2*k)+factorial(k)+1,k=0..600)]); # Muniru A Asiru, May 27 2018

lista(nn) = {for(k=0, nn, if(ispseudoprime(p=(2*k)!+k!+1), print1(p, ", ")));} \\ Altug Alkan, Mar 22 2018

a(n) = (2*A242487(n))! + A242487(n)! + 1.

A062701 Index of factorial primes of the form k! + 1.

Original entry on oeis.org

1, 2, 4, 2428957

Offset: 1

Labos Elemer, Jul 11 2001

			The exact subscript of the 5th prime [1 + 27! = 10888869450418352160768000001] is not yet available.

Cf. A000720, A002981, A002982, A055490, A088332.

a(n) = PrimePi(A002981(n)!+1).

Offset 1 from Michel Marcus, Aug 29 2019

A093437 a(n) = largest prime of the form n!/k! + 1.

Original entry on oeis.org

2, 2, 3, 7, 13, 61, 31, 2521, 20161, 15121, 604801, 39916801, 3991681, 3113510401, 14529715201, 54486432001, 10461394944001, 59281238016001, 53353114214401, 2, 670442572801, 8515157028618240001, 9366672731480064001

Offset: 0

Amarnath Murthy, Apr 01 2004

nonn

Is 19 the largest n such that a(n) = 2? There are none for 19 < n <= 600. - Robert Israel, Jan 16 2017

			a(7) = 2521 because 7!/2! + 1 = 2521 is prime, whereas 7!/1! + 1 = 5041 = 71^2 is composite;
a(19) = 2 because the only prime of the form 19!/k! + 1 is 19!/19! + 1 = 2.

Robert Israel, Table of n, a(n) for n = 0..466

Cf. A093621 (smallest k > 0 such that n!/k! + 1 is prime), A002981 (n! + 1 is prime), A088332 (primes of form n! + 1).

f:= proc(n) local k,x;
  x:= n!;
  for k from 2 do
    if isprime(x+1) then return x+1 fi;
    x:= x/k;
  od
end proc:
map(f, [$0..40]); # Robert Israel, Jan 16 2017

a[n_] := Module[{k, x}, x = n!; For[k = 2, True, k++, If[PrimeQ[x+1], Return[x+1]]; x = x/k]];
Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Feb 08 2023, after Robert Israel *)

Corrected and extended by Hugo Pfoertner, Apr 06 2004

A103319 Primes of the form p! + 1 where p is prime.

Original entry on oeis.org

3, 7, 39916801, 13763753091226345046315979581580902400000001, 33452526613163807108170062053440751665152000000001, 4470115461512684340891257138125051110076800700282905015819080092370422104067183317016903680000000000000001

Offset: 1

Jonathan Sondow, Jan 31 2005

The values of p are 2, 3, 11, 37, 41, 73 which is A093804 (with a different definition). Subsequence of A088332 (primes of the form n! + 1).

			2 and 2! + 1 = 3 are prime, so 3 is a member.

R. K. Guy, Unsolved Problems in Number Theory, Section A2.

R. Mestrovic, Euclid's theorem on the infinitude of primes: a historical survey of its proofs (300 BC--2012) and another new proof, arXiv preprint arXiv:1202.3670, 2012 - From N. J. A. Sloane, Jun 13 2012
Eric Weisstein's World of Mathematics, Factorial Prime
Index entries for sequences related to factorial numbers.

Cf. A093804, A002981, A088332, A103318.

Select[Table[p!+1,{p,Prime[Range[30]]}],PrimeQ] (* Harvey P. Dale, Nov 28 2019 *)

A084727 Primes arising in A084726.

Original entry on oeis.org

2, 3, 7, 281, 76561, 576577, 17873857, 643458817, 337767408001, 21617114112001, 39916801, 119715577952256001, 1980990543353657472001, 26582634158080001, 3577861898239093446857008573440001, 711975497511453268455460274177

Offset: 1

Amarnath Murthy and Meenakshi Srikanth (menakan_s(AT)yahoo.com), Jun 13 2003

nonn

Smallest prime of the form: 1 + product of n terms of an arithmetic progression with first term 1.
Conjecture: All terms exist.
If n! + 1 is prime (A002981) then a(n) = A088332(n). - Hugo Pfoertner, Nov 18 2004

			a(1) = 2 = 1 + 1;
a(4) = 281 = 1*4*7*10 + 1 (1*2*3*4 + 1 = 25 is composite);
a(5) = 76561 = 1*8*15*22*29 + 1.

Nathaniel Johnston, Table of n, a(n) for n = 1..100

Cf. A002981, A084726, A088332.

A084727 := proc(n) local k,p: for k from 1 do p:=1+mul(1+j*k,j=0..n-1): if(isprime(p))then return p: fi: od: end: seq(A084727(n),n=1..16); # Nathaniel Johnston, Jun 26 2011

np[n_]:=Module[{k=1},While[!PrimeQ[Times@@NestList[k+#&,1,n-1]+1],k++];Times@@NestList[k+#&,1,n-1]+1]; Array[np,20] (* Harvey P. Dale, Aug 05 2021 *)

a(n) = 1 + Product_{i = 0..n-1} (1 + i*A084726(n)). - David Wasserman, Jan 03 2005

More terms from David Wasserman, Jan 03 2005

A181764 Numbers n such that n!+1 is a product of two distinct prime numbers.

Original entry on oeis.org

6, 8, 10, 13, 14, 19, 20, 24, 25, 26, 28, 34, 38, 48, 54, 55, 59, 71, 75, 92, 109, 114, 115

Offset: 1

Vladimir Joseph Stephan Orlovsky, Nov 13 2010

n! + 1 must be the product of two distinct prime numbers and also the product of only two prime numbers counted with multiplicity. Thus, 12 is NOT a term of the sequence because 12! + 1 = 13*13*2834329. - Harvey P. Dale, Jul 22 2019

Other terms in this sequence: 392, 551, 601, 770, 772, 878, 1033, 1320, 1831, 2620, 2808, 3752, 4233, 4616, 4984, 7260. - Chai Wah Wu, Feb 28 2020

			6!+1=7*103; 8!+1=61*661; 10!+1=11*329891; 13!+1=83*75024347; 14!+1=23*3790360487; 19!+1=71*1713311273363831;..

Bruce C. Berndt and William F. Galway, On the Brocard-Ramanujan diophantine equation n!+1=m^2, The Ramanujan Journal, March 2000, Volume 4, Issue 1, pp 41-42.

Cf. A033312, A038507, A078781, A085692, A085747, A088332.

Subsequence of A078778.

fQ[n_]:=Last/@FactorInteger[n]=={1,1}; Select[Range[40], fQ[#!+1]&]

Extended by D. S. McNeil, Nov 13 2010
One more term (114) (factored by Womack et al.) from Sean A. Irvine, May 25 2015
One more term (115) (factored by Womack et al.) from Sean A. Irvine, Feb 08 2016

A301523 Integers which can be partitioned into two distinct factorials. 0! and 1! are not considered distinct.

Original entry on oeis.org

3, 7, 8, 25, 26, 30, 121, 122, 126, 144, 721, 722, 726, 744, 840, 5041, 5042, 5046, 5064, 5160, 5760, 40321, 40322, 40326, 40344, 40440, 41040, 45360, 362881, 362882, 362886, 362904, 363000, 363600, 367920, 403200, 3628801, 3628802, 3628806, 3628824, 3628920, 3629520, 3633840, 3669120, 3991680

Offset: 1

Seiichi Manyama, Mar 23 2018

nonn

Numbers of the form i! + j! where i > j > 0. - Altug Alkan, Mar 23 2018

Primes in this sequence are A088332(n) for n > 1.

			    + |   1    2    6   24
  ----+--------------------
    1 |
    2 |   3;
    6 |   7,   8;
   24 |  25,  26,  30;
  120 | 121, 122, 126, 144;

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A000142, A001048, A038507, A059590, A088332, A301593.

Union[Total/@Subsets[Range[10]!,{2}]] (* Harvey P. Dale, Aug 25 2020 *)

A123909 Primes of the form k!+11.

Original entry on oeis.org

13, 17, 131, 5051, 3628811

Offset: 1

Alexander Adamchuk, Oct 28 2006

Corresponding numbers k such that k!+11 is prime are {2, 3, 5, 7, 10}.

Cf. A088332 - Primes of the form n!+1.

Select[Table[n!+11,{n,1,1000}],PrimeQ[ # ]&]

cp's OEIS Frontend

A118370 Divisorial primes: Primes p such that p = 1 + Product_{d|n} d for some n (ordered by n).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

A300947 Primes of form (2*k)! + k! + 1.

Views

Author

Keywords

Comments

Crossrefs

Programs

Maple

PARI

Formula

A062701 Index of factorial primes of the form k! + 1.

Views

Author

Keywords

Examples

Crossrefs

Formula

Extensions

A093437 a(n) = largest prime of the form n!/k! + 1.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Extensions

A103319 Primes of the form p! + 1 where p is prime.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

A084727 Primes arising in A084726.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

Extensions

A181764 Numbers n such that n!+1 is a product of two distinct prime numbers.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Extensions

A301523 Integers which can be partitioned into two distinct factorials. 0! and 1! are not considered distinct.

Views

Author