A078778 -id:A078778 - cp's OEIS frontend

A085747 Least positive k such that n! + k is a semiprime.

3, 2, 3, 1, 1, 1, 1, 1, 5, 1, 3, 17, 1, 1, 7, 2, 3, 23, 1, 1, 11, 29, 3, 1, 1, 1, 37, 1, 41, 2, 19, 11, 11, 1, 7, 3, 41, 1, 13, 127, 47, 59, 2, 37, 5, 37, 59, 1, 2, 73, 59, 79, 73, 1, 1, 61, 118, 37, 1, 61, 31, 37, 43, 7, 83, 71, 19, 7, 103, 19, 1, 107, 37, 167, 1, 103, 3, 73, 109, 97

Offset: 1

Jason Earls, Jul 21 2003

a(106) >= 139. - Sean A. Irvine, Oct 29 2019

a(n) = 1 iff n in A078778.

Tyler Busby, Table of n, a(n) for n = 1..107 (terms 1..99, 101..103, and 105 from Max Alekseyev, terms 100 and 104 from Sean A. Irvine)
Dario Alejandro Alpern, Factorization using the Elliptic Curve Method
factordb, Status of 106!+139.
Hisanori Mishima, Factorization of n! + 1 (n = 1 to 100)

Cf. A089538, A089539, A089540, A089541.

m:=1; sol:=[]; for n in [1..40] do k:=1; while &+[d[2]: d in Factorization(Factorial(n)+k)] ne 2 do k:=k+1; end while; sol[m]:=k; m:=m+1; end for; sol; // Marius A. Burtea, Aug 24 2019

Table[SelectFirst[Range@ 1000, PrimeOmega[n! + #] == 2 &], {n, 40}] (* Michael De Vlieger, Mar 08 2016, Version 10 *)
lpk[n_]:=Module[{k=1,fn=n!},While[PrimeOmega[fn+k]!=2,k++];k]; Array[lpk,80] (* Harvey P. Dale, Feb 18 2025 *)

a(n) = {k = 1; while (bigomega(n!+k) != 2, k++); k;} \\ Michel Marcus, Mar 08 2016

Extended by Robert G. Wilson v, Jul 27 2003
a(55) from Ray Chandler, Nov 09 2003
a(56)-a(80) from Sean A. Irvine, Mar 29 2010

A078781 Numbers n such that n!-1 is a semiprime.

Original entry on oeis.org

5, 8, 10, 13, 16, 20, 23, 24, 26, 27, 34, 36, 40, 47, 50, 59, 68, 79, 85, 93, 137, 143, 151

Offset: 1

Jason Earls, Jan 09 2003

The next candidate for a continuation is 154!-1, which is composite with 272 decimal digits and unknown factorization. Further known terms are 157, 229, 381, 390, 392, 400, 814, 929; factorization unknown for 154, 196, 232, 271, 307, 322, 332, 333, 334, 350, 352, 386, 389, 443, 449, ...
Note that the two prime factors of 24!-1 = 620448401733239439359999 = 625793187653 * 991459181683 both have 12 decimal digits.
There is another term with prime factors with equal number of decimal digits: 34!-1 = 10398560889846739639*28391697867333973241 (20 digits each)
From Antti Karttunen, Dec 27 2015: (Start)
Furthermore, both factors of 24!-1 are in binary system 40 bits long (A070939), and in factorial base representation (A007623) they both have 14 digits: <7,2,6,5,4,8,2,3,0,0,2,0,2,1> and <11,5,2,10,1,5,6,3,4,1,1,3,0,1>. That is, A007623(625793187653) = 72654823002021, but the latter number cannot be represented reliably in such a more compact form, because it already contains digits > 9.
Factors of 34!-1 are 64 and 65 bits long, and their factorial base representations contain both 20 digits: <4,5,9,3,1,13,11,7,9,1,0,6,1,1,6,5,3,1,0,1> and  <11,13,7,10,0,12,3,4,6,11,1,8,1,4,2,2,1,2,2,1>.
Also the factors of 5!-1 = 119 = 7*17 are both of the same length in factorial base system: "101" and "221".
(End)
1338, 1447, 1788, 1824, 2805, 2881, 2960, 5824 are also terms of the sequence. - Chai Wah Wu, Feb 28 2020

FactorDB, Status of 154!-1.
Paul Leyland, Tables of factors of N!+1 and N!-1.
Andrew Walker, Factors of n!-1 for n>=400.

Cf. A001358, A080802.
Cf. A078778 (numbers such that n!+1 is a semiprime).
Cf. also A007623, A070939, A266344.

IsSemiprime:=func< n | &+[ k[2]: k in Factorization(n) ] eq 2 >; [n: n in [3..50] | IsSemiprime(Factorial(n)-1)]; // Vincenzo Librandi, Dec 28 2015

Select[Range[50], PrimeOmega[#! - 1] == 2 &] (* Vincenzo Librandi, Dec 28 2015 *)

{ fm(a,b)=local(c,d,r); for(n=a,b,r=n!-1; c=vecmin(factor(r)[,1]~); d=vecmax(factor(r)[,1]~); if(bigomega(r)==2 && isprime(c) && isprime(d), print1(n" ");))} fp(2,100)

More terms from Hugo Pfoertner, Apr 05 2003

a(23) added by Daniel Suteu, Mar 30 2019

A064144 a(n) is the number of divisors of n!+1.

Original entry on oeis.org

2, 2, 2, 3, 3, 4, 3, 4, 8, 4, 2, 6, 4, 4, 8, 32, 8, 64, 4, 4, 8, 8, 12, 4, 4, 4, 2, 4, 8, 32, 16, 16, 32, 4, 32, 64, 2, 4, 16, 128, 2, 8, 16, 8, 8, 8, 16, 4, 32, 32, 64, 16, 16, 4, 4, 16, 8, 16, 4, 16, 16, 8, 32, 8

Offset: 1

Vladeta Jovovic, Sep 11 2001

nonn

Max Alekseyev, Table of n, a(n) for n = 1..139
William Gerst, A conjecture on the prime factorization of n!+1, arXiv:1809.07360 [math.GM], 2018.
Hisanori Mishima, Factorizations of many number sequences

Cf. A000005, A002583, A027423, A002981, A002982, A064145, A078778, A181764.

Do[ Print[ DivisorSigma[0, n! + 1]], {n, 1, 40} ]

a(n) = numdiv(n! + 1); \\ Harry J. Smith, Sep 09 2009

from math import factorial
from sympy import divisor_count
def A064144(n): return divisor_count(factorial(n)+1) # Chai Wah Wu, Oct 20 2023

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

More terms from Robert G. Wilson v, Oct 04 2001
a(42)-a(64) from Harry J. Smith, Sep 09 2009
Edited by Jon E. Schoenfield, Jun 21 2018

A082952 Smaller of the two factors of the n-th semiprime number of the form m!+1.

Original entry on oeis.org

5, 11, 7, 71, 61, 11, 83, 23, 71, 20639383, 811, 401, 1697, 29, 67411, 14029308060317546154181, 12893, 12318573951317236818169524329, 79, 16567, 6653, 293, 229758023927584562777368125832724248866067995638905559798117

Offset: 1

Hugo Pfoertner, May 26 2003

nonn

			a(3)=7 because A078778(3)!+1=6!+1=721=7*103

Paul Leyland, Tables of factors of N!+1 and N!-1.
Hisanori Mishima, n! + 1 (n = 1 to 100).

Cf. A046315, A078778, A002981, A080802.

FactorInteger[#][[1,1]]&/@Select[Range[50]!+1,PrimeOmega[#]==2&] (* The program generates the first 17 terms of the sequence. To generate more, increase the Range constant but the program may take a long time to run.*) (* Harvey P. Dale, Dec 11 2023 *)

Numbers p such that p*q=A078778(n)!+1, p, q prime, p

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

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

A250293 Primes p such that p#+1 is a semiprime, where # is the primorial (A034386).

Original entry on oeis.org

13, 19, 23, 43, 61, 67, 73, 83, 101, 139, 151, 173, 223, 251, 383, 457, 571, 673, 761

Offset: 1

Eric Chen, Dec 24 2014

The next candidate after 571 is 859. 859# + 1 is a 359-digit composite with no known factors. - Hugo Pfoertner, Feb 05 2021

			a(1) = 13 so 13# + 1 = 30031 = 59 * 509 is a semiprime.

factordb.com, Status of 859#+1.
Hisanori Mishima, Factorizations of many number sequences, PI Pn + 1 (n = 1 to 110) (2012).

Cf. A005234, A006862, A034386, A078778, A085725, A250294.

Prime[#]&/@(Module[{pmrl=FoldList[Times,Prime[Range[50]]]},Position[ pmrl, ?(PrimeOmega[ #+1]==2&)]]//Flatten) (* _Harvey P. Dale, Apr 29 2017 *)

a(n) = prime(A085725(n)). - Hugo Pfoertner, Feb 05 2021

a(16)-a(18) using factordb.com from Hugo Pfoertner, Feb 05 2021

Missing 571 inserted by Sean A. Irvine, Mar 03 2023

A098594 Numbers n such that n!-1 and n!+1 are both semiprime.

Original entry on oeis.org

5, 8, 10, 13, 20, 24, 26, 34, 59, 392

Offset: 1

Mark Hudson (mrmarkhudson(AT)hotmail.com), Sep 17 2004

This sequence is formed of all those terms that appear in both A078778 and A078781.

a(11) >= 929. 929!-1 is semiprime, no factor of 929!+1 is known. - Sean A. Irvine, Mar 09 2013

			10!+1 = 3628801 = 11*329891 and 10!-1 = 3628799 = 29*125131 so 10 is a member of the sequence.
464 is not a term since 464!-1=2828197538205421590987128183441789966021011*C996 is not a semiprime. - _Sean A. Irvine_, Mar 09 2013

Cf. A078778, A078781.

out:=[]: for n from 1 to 60 do: a:=n!-1: b:=n!+1: if (bigomega(a)=2) and (bigomega(b)=2) then out:=[op(out),n]: print(n): fi: od: out;

Select[Range[35],PrimeOmega[#!+{1,-1}]=={2,2}&] (* The program generates the first 8 terms of the sequence. To generate more, increase the Range constant but the program may take a long time to run. *) (* Harvey P. Dale, Aug 13 2023 *)

a(10) from D. S. McNeil, Sep 04 2011

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A085747 Least positive k such that n! + k is a semiprime.

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A078781 Numbers n such that n!-1 is a semiprime.

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A064144 a(n) is the number of divisors of n!+1.

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A082952 Smaller of the two factors of the n-th semiprime number of the form m!+1.

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A090159 Semiprimes of the form m! + 1.

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A090160 Greater prime factor of semiprimes in A090159.

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

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A250293 Primes p such that p#+1 is a semiprime, where # is the primorial (A034386).

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