A038507 -id:A038507 - cp's OEIS frontend

A037152 Smallest prime > n!+1.

3, 5, 11, 29, 127, 727, 5051, 40343, 362897, 3628811, 39916817, 479001629, 6227020867, 87178291219, 1307674368043, 20922789888023, 355687428096031, 6402373705728037, 121645100408832089, 2432902008176640029

Offset: 1

Jud McCranie

nonn

Main entry for this sequence is A037153.

Vincenzo Librandi, Table of n, a(n) for n = 1..100

Cf. A000142, A037153, A038507, A151800.

NextPrime[Range[20]!+1] (* Harvey P. Dale, Apr 08 2012 *)

makelist(next_prime(n!+1), n,  1, 20); /* Bruno Berselli, May 20 2011 */

for(n=1,100,print1(nextprime(n!+2),", ")); /* Joerg Arndt, May 21 2011 */

a(n) = A151800(A000142(n)+1) = A000142(n) + A037153(n).

a(n) = A151800(A038507(n)). - Michel Marcus, Feb 18 2024

Extended by Ray Chandler, Mar 07 2010

A056111 Highest proper factor of n!+1.

Original entry on oeis.org

1, 1, 1, 1, 5, 11, 103, 71, 661, 19099, 329891, 1, 36846277, 75024347, 3790360487, 22163972339, 1230752346353, 538105034941, 336967037143579, 1713311273363831, 117876683047, 1188161445853707907, 48869596859895986087, 550042909337978226383, 765041185860961084291

Offset: 0

Henry Bottomley, Jun 12 2000

nonn

Amiram Eldar, Table of n, a(n) for n = 0..139
Hisanori Mishima, Factorizations of many number sequences
Hisanori Mishima, Factorizations of many number sequences
R. G. Wilson v, Explicit factorizations

Cf. A002583, A038507, A051301, A056110.

Mathematica
```
a[n_] := Divisors[n!+1][[ -2]]
```

a(n) = A038507(n)/A051301(n).

Corrected and extended by Dean Hickerson, Aug 30 2001

More terms from Amiram Eldar, Oct 07 2019

A073829 a(n) = 4*((n-1)! + 1) + n.

Original entry on oeis.org

9, 10, 15, 32, 105, 490, 2891, 20172, 161293, 1451534, 14515215, 159667216, 1916006417, 24908083218, 348713164819, 5230697472020, 83691159552021, 1422749712384022, 25609494822912023, 486580401635328024, 9731608032706560025, 204363768686837760026

Offset: 1

Reinhard Zumkeller, Aug 12 2002

Paulo Ribenboim, The Little Book of Bigger Primes, Springer-Verlag NY 2004. See p. 192.
James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 112.

Winston de Greef, Table of n, a(n) for n = 1..449 (first 200 terms from Vincenzo Librandi)
P. A. Clement, Congruences for sets of primes, Am. Math. Monthly 56 (1) (1949) 23-25.

See A073830 for motivation.

Cf. A038507.

[4*(Factorial(n-1)+1)+n: n in [1..20]]; // Vincenzo Librandi, May 04 2014

Table[(4 ((n - 1)! + 1) + n), {n, 1, 20}] (* Vincenzo Librandi, May 04 2014 *)

[4*(factorial(n-1) + 1) + n for n in range(1,22)] # Stefano Spezia, Apr 21 2025

A090159 Semiprimes of the form m! + 1.

Original entry on oeis.org

25, 121, 721, 5041, 40321, 3628801, 6227020801, 87178291201, 121645100408832001, 2432902008176640001, 620448401733239439360001, 15511210043330985984000001, 403291461126605635584000001, 304888344611713860501504000001, 295232799039604140847618609643520000001

Offset: 1

Ray Chandler, Nov 22 2003

nonn

Cf. A038507, A078778, A082952, A090160.

a(n) = A078778(n)! + 1.

Offset changed to 1 and more terms from Jinyuan Wang, Jul 31 2021

A356668 Expansion of e.g.f. Sum_{k>=0} x^k / (k! - k*x^k).

Original entry on oeis.org

1, 1, 3, 7, 37, 121, 1141, 5041, 60761, 378001, 5444461, 39916801, 729041545, 6227020801, 130767460825, 1321314894901, 31388220966961, 355687428096001, 9636906872926477, 121645100408832001, 3649432697160095561, 51223991519836175041, 1686001091666419279753

Offset: 0

Seiichi Manyama, Aug 22 2022

nonn

Cf. A356029, A356328, A356608.

Cf. A038507, A327578, A356667.

a[n_]:= n! * DivisorSum[n, 1/(# * (# - 1)!^(n/#)) &]; a[0] = 1; Array[a, 23, 0] (* Amiram Eldar, Aug 22 2022 *)

my(N=30, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, x^k/(k!-k*x^k))))

a(n) = if(n==0, 1, n!*sumdiv(n, d, 1/(d*(d-1)!^(n/d))));

Expansion of e.g.f. Sum_{k>=0} x^k / (k! * (1 - k*x^k/k!)).
a(n) = n! * Sum_{d|n} 1/(d * (d-1)!^(n/d)) for n > 0.
a(p) = 1 + p! for prime p.

A358389 a(n) = n * Sum_{d|n} (d + n/d - 2)!/d!.

Original entry on oeis.org

1, 3, 7, 29, 121, 745, 5041, 40425, 362917, 3629411, 39916801, 479006233, 6227020801, 87178326495, 1307674369891, 20922790211057, 355687428096001, 6402373709009185, 121645100408832001, 2432902008212933061, 51090942171709581289, 1124000727778046764823

Offset: 1

Seiichi Manyama, Nov 13 2022

Michael De Vlieger, Table of n, a(n) for n = 1..449

Cf. A038507, A343573.

Table[n*DivisorSum[n, ((# + n/# - 2)!)/(#!) &], {n, 22}] (* Michael De Vlieger, Nov 13 2022 *)

PARI
```
a(n) = n*sumdiv(n, d, (d+n/d-2)!/d!);
```

my(N=30, x='x+O('x^N)); Vec(sum(k=1, N, k!*(x/(1-x^k))^k))

G.f.: Sum_{k>0} k! * (x/(1 - x^k))^k.

If p is prime, a(p) = 1 + p!.

A090160 Greater prime factor of semiprimes in A090159.

Original entry on oeis.org

5, 11, 103, 71, 661, 329891, 75024347, 3790360487, 1713311273363831, 117876683047, 765041185860961084291, 38681321803817920159601, 237649652991517758152033, 10513391193507374500051862069, 4379593820587205958191075783529691, 37280713718589679646221

Offset: 1

Ray Chandler, Nov 22 2003

nonn

Cf. A038507, A078778, A082952, A090159.

Offset changed to 1 and more terms from Jinyuan Wang, Aug 01 2021

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

A213169 a(n) = n! + n + 1.

Original entry on oeis.org

2, 3, 5, 10, 29, 126, 727, 5048, 40329, 362890, 3628811, 39916812, 479001613, 6227020814, 87178291215, 1307674368016, 20922789888017, 355687428096018, 6402373705728019, 121645100408832020, 2432902008176640021

Offset: 0

Olivier Gérard, Nov 02 2012

Can be used to detect triangular tables whose rows sum to n! when decreased by 1.

Vincenzo Librandi, Table of n, a(n) for n = 0..200
R. C. Castillo, On the Sum of Corresponding Factorials and Triangular Numbers: Runsums, Trapezoids and Politeness, Asia Pacific Journal of Multidisciplinary Research, 3 (2015), 95-101.
Aria Chen, Tyler Cummins, Rishi De Francesco, Jate Greene, Tanya Khovanova, Alexander Meng, Tanish Parida, Anirudh Pulugurtha, Anand Swaroop, and Samuel Tsui, Card Tricks and Information, arXiv:2405.21007 [math.HO], 2024. See p. 5.

Cf. A000142, A038507, A005095.

Mathematica
```
Table[n! + n + 1, {n, 0, 20}]
```

A213169(n):=n!+n+1$
makelist(A213169(n),n,0,30); /* Martin Ettl, Nov 03 2012 */

A227546 a(n) = n! + n^2 + 1.

Original entry on oeis.org

2, 3, 7, 16, 41, 146, 757, 5090, 40385, 362962, 3628901, 39916922, 479001745, 6227020970, 87178291397, 1307674368226, 20922789888257, 355687428096290, 6402373705728325, 121645100408832362, 2432902008176640401, 51090942171709440442, 1124000727777607680485

Offset: 0

Vincenzo Librandi, Jul 26 2013

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Cf. A000142, A005095, A038507, A213169, A033312, A005096, A004664.

Cf. A119662 (primes of the form k! + k^2 + 1).

Magma
```
[Factorial(n)+n^2+1: n in [0..25]];
```
Mathematica
```
Table[n! + n^2 + 1, {n, 0, 30}]
```

/* By the recurrence: */ a[0]:2$ a[1]:3$ a[n]:=(n^4-5*n^3+8*n^2-5*n-1)*a[n-1]/(n^3-6*n^2+11*n -7)-(n-1)*(n^3-3*n^2+2*n-1)*a[n-2]/(n^3-6*n^2+11*n-7)$ makelist(a[n], n, 0, 21); /* Bruno Berselli, Jul 26 2013 */

(n^3 -6*n^2 +11*n -7)*a(n) -(n^4 -5*n^3 +8*n^2 -5*n -1)*a(n-1) +(n-1)*(n^3 -3*n^2 +2*n -1)*a(n-2) = 0 for n>1. - Bruno Berselli, Jul 26 2013

cp's OEIS Frontend

A037152 Smallest prime > n!+1.

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A056111 Highest proper factor of n!+1.

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A073829 a(n) = 4*((n-1)! + 1) + n.

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Sage

A090159 Semiprimes of the form m! + 1.

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A356668 Expansion of e.g.f. Sum_{k>=0} x^k / (k! - k*x^k).

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Mathematica

PARI

PARI

Formula

A358389 a(n) = n * Sum_{d|n} (d + n/d - 2)!/d!.

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Mathematica

PARI

PARI

Formula

A090160 Greater prime factor of semiprimes in A090159.

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Extensions

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

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Mathematica

Extensions