A078781 -id:A078781 - cp's OEIS frontend

A078778 Numbers n such that n!+1 is a semiprime.

4, 5, 6, 7, 8, 10, 13, 14, 19, 20, 24, 25, 26, 28, 34, 38, 48, 54, 55, 59, 71, 75, 92, 109, 114, 115

Offset: 1

Jason Earls, Jan 09 2003

Subsequence of (and likely is equal to) the union of A146968 and A181764. - Max Alekseyev, May 28 2015
Note that the two prime factors of 38!+1 = 523022617466601111760007224100074291200000001 = 14029308060317546154181 * 37280713718589679646221 both have 23 decimal digits. Are there any other terms in this sequence other than 4,5,7 and 38 with this property?
a(27) > 139. - Robert Price, Apr 11 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

			4 is in the sequence because 4!+1=25=5*5 is semiprime. But 9 is not in the sequence because 9!+1=19*71*269 is not semiprime. - _Sean A. Irvine_, Nov 15 2009

FactorDB, Status of 140!+1.

Cf. A001358, A082952, A090159, A090160, A078781.

IsSemiprime:=func< n | &+[ k[2]: k in Factorization(n) ] eq 2 >; [n: n in [1..60] | IsSemiprime(Factorial(n)+1)]; // Vincenzo Librandi, May 26 2015

Select[Range[100], Plus@@Last/@FactorInteger[#! + 1]==2 &] (* Vincenzo Librandi, May 26 2015 *)
Select[Range[100],PrimeOmega[#!+1]==2&] (* Harvey P. Dale, Mar 19 2017 *)

{ fp(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(1,100)

Term 109 from Sean A. Irvine, Nov 15 2009
Term 114 (factored by Womack et al.) from Sean A. Irvine, May 25 2015
Term 115 (factored by Womack et al.) from Sean A. Irvine, Feb 08 2016

A266344 a(n) = number of ways n can be divided into two factors that have the same number of digits in factorial base representation (the two different orders for unequal factors are counted only once).

Original entry on oeis.org

1, 0, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 0, 0, 1, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 1, 1, 0, 1, 0, 0, 0, 1, 0, 2, 0, 0, 0, 0, 1, 1, 0, 1, 1, 0, 0, 2, 0, 0, 0, 1, 0, 2, 1, 0, 0, 0, 0, 2, 0, 1, 1, 1, 0, 1, 0, 1, 1, 0, 0, 2, 0, 1, 0, 2, 0, 1, 0, 0, 1, 0, 1, 3

Offset: 1

Antti Karttunen, Dec 28 2015

			1 can be factored just one way, as 1*1, and thus a(1) = 1.
4 can be factored as 2*2, and thus also a(4) = 1, and generally for all perfect squares k, a(k) >= 1.
14 can be factored as 2*7, but as A007623(2) = 2 and A007623(7) = 101, with different number of digits in factorial base (and 1*14 fares even less well), a(14) = 0.
72 can be factored to two divisors so that the factorial base representations are of equal length as 6*12 or 8*9 (where the corresponding factorial base representations are "100" * "200" and "110" * "111"), thus a(72) = 2.
120 can be similarly factored as 6*20 ("100" * "310"), 8*15 ("110" * "211") and 10*12 ("120" * "200"), thus a(120) = 3.

Antti Karttunen, Table of n, a(n) for n = 1..14161
Index entries for sequences related to factorial base representation

Cf. A084558.
Cf. A266345 (positions of records).
Cf. also A078781, A266342.

r = Most@ Reverse@ Range@ 10; Map[Length, Table[Flatten@ Map[Differences@ IntegerLength[#, MixedRadix@ r] &, Transpose@ {#, n/#}] &@ TakeWhile[Divisors@ n, # <= Sqrt@ n &], {n, 120}] /. k_ /; k > 0 -> Nothing] (* Michael De Vlieger, Dec 30 2015, Version 10.2 *)

A084558(n) = { my(m=1); if(0==n,n,while(m!<=n,m++);return(m-1)); }
A266344(n) = sumdiv(n, d, ((d <= (n/d)) && (A084558(d)==A084558(n/d))));
for(n=1, 14161, write("b266344.txt", n, " ", A266344(n)));

a(n) = Sum_{d|n} [(d <= (n/d)) and (A084558(d) = A084558(n/d))].

(In the above formula [ ] stands for Iverson bracket, giving as its result 1 only if d is less than or equal to n/d and in factorial base representation d and n/d require equal number of digits, and 0 otherwise.)

A080802 Smaller of the two factors of the n-th semiprime number of the form m!-1.

Original entry on oeis.org

7, 23, 29, 1733, 3041, 124769, 51871, 625793187653, 20431, 29, 10398560889846739639, 155166770881, 9190813196017748117, 340777, 3282689, 61, 895269581, 21289796287569735866708594882309656982337071, 14380211646881467415803462581621417951534002839, 884057, 139, 7533609175373352257, 1712114014849097863989021395568379341467597467171639484099

Offset: 1

Hugo Pfoertner, Mar 25 2003

To continue the sequence the factorizations of 151!-1 and 154!-1 are required, which are composite numbers with 265 and 272 digits, respectively. The next term would then be 37272934189201737869016720929 (factor of 157!-1).

151!-1 has been factored into P58*P208. - Hugo Pfoertner, Jul 18 2019

			a(1)=7 because A078781(1)!-1=5!-1=7*17,
a(2)=23 because A078781(2)!-1=8!-1=23*1753,
a(11)=10398560889846739639 because A078781(11)!-1=34!-1= 10398560889846739639*28391697867333973241 (20 digits each).

Dario Alejandro Alpern, Factorization using the Elliptic Curve Method
Paul Leyland, Factor table Updated 12 May 2007.
Hisanori Mishima, Factors of n! - 1

Cf. A078781.

Numbers p such that p*q=A078781(n)!-1, p, q prime, p

a(23) from Hugo Pfoertner, Jul 18 2019

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

A250294 Primes p such that p#-1 is a semiprime, where # is the primorial (A034386).

Original entry on oeis.org

7, 17, 29, 31, 43, 59, 71, 73, 97, 101, 223, 233, 257, 439, 503, 709, 859, 863, 1013

Offset: 1

Eric Chen, Dec 24 2014

1153 and 1381 are also terms. - Amiram Eldar, Feb 16 2020

a(20) >= 1091. 1091# - 1 is a 458-digit composite with no known factors. - Hugo Pfoertner, Feb 05 2021

			a(2) = 17 so 17# - 1 = 510509 = 61 * 8369 is a semiprime.

FactorDB Factorization of 1013#-1.
FactorDB Status of 1091#-1.
Hisanori Mishima, Factorizations of many number sequences, PI Pn - 1 (n = 1 to 110) (2013).

Cf. A001221, A001358, A002110, A006794, A034386, A057588, A078781, A250293.

p = Select[Range[101], PrimeQ]; p[[ Position[FoldList[Times, p] - 1, ?(PrimeOmega[#] == 2 &)] //Flatten ]] (* _Amiram Eldar, Feb 16 2020 *)

A001221(A034386(a(n)) - 1) = 2. - Amiram Eldar, Feb 16 2020

a(15)-a(18) using factordb.com from Amiram Eldar, Feb 16 2020
a(19) using factordb.com from Hugo Pfoertner, Feb 05 2021
Edited by Max Alekseyev, Aug 26 2021

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

Showing 1-6 of 6 results.

cp's OEIS Frontend

A078778 Numbers n such that n!+1 is a semiprime.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Extensions

A266344 a(n) = number of ways n can be divided into two factors that have the same number of digits in factorial base representation (the two different orders for unequal factors are counted only once).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A080802 Smaller of the two factors of the n-th semiprime number of the form m!-1.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

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

A250294 Primes p such that p#-1 is a semiprime, where # is the primorial (A034386).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

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

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Maple

Mathematica

Extensions