A051301 -id:A051301 - cp's OEIS frontend

A160245 a(n) = index of the n-th prime in A051301 (least prime factor of m!+1).

2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 1, 1, 2, 2, 1, 3, 2, 2, 6, 1, 3, 3, 1, 1, 1, 1, 1, 2, 1, 1, 2, 3, 1, 2, 1, 1, 2, 1, 1, 2, 1, 2, 1, 1, 2, 1, 1, 4, 1, 2, 1, 1, 3, 3, 2, 2, 3, 1, 1, 1, 5, 3, 2, 1, 2, 1, 1, 2, 1, 1, 2, 2, 1, 2, 4, 2, 2, 5, 1, 2, 1, 2, 1, 1, 2, 2, 1, 2, 3, 3, 3, 1, 2, 1, 3, 1, 1, 2, 1, 1, 2, 4, 2, 4

Offset: 1

Frederick Magata (frederick.magata(AT)web.de), May 05 2009

nonn

Because of Wilson's theorem A051301(p-1)=p for every prime p. Hence a(n)>0, and since A051301(k)>k, a(n) is actually finite.

The first 18 values of the sequence were calculated with Maple. The others were derived from T. D. Noe's b-file for b051301.txt.

			a(17)=3 because A051301(15)=A051301(43)=A051301(58)=59, and there are no other occurrences of 59=17th prime number in A051301.

Cf. A051301, A115092.

a:=proc(n) option remember; local k,l,p: p:=ithprime(n): l:=0: for k from 0 to p-2 do if A051301(k)=p then l:=l+1; fi; od; l+1; end;

prev={}; Table[p=Prime[n]; s=Select[Complement[Range[0,p-1],prev], Mod[ #!+1,p]==0&]; prev=Union[s,prev]; Length[s], {n,100}] (* T. D. Noe, May 12 2009 *)

Extended by T. D. Noe, May 12 2009

A038507 a(n) = n! + 1.

Original entry on oeis.org

2, 2, 3, 7, 25, 121, 721, 5041, 40321, 362881, 3628801, 39916801, 479001601, 6227020801, 87178291201, 1307674368001, 20922789888001, 355687428096001, 6402373705728001, 121645100408832001, 2432902008176640001, 51090942171709440001, 1124000727777607680001, 25852016738884976640001

Offset: 0

N. J. A. Sloane

"For n = 4, 5 and 7, n!+1 is a square. Sierpiński asked if there are any other values of n with this property." p. 82 of Ogilvy and Anderson (see A146968).
Number of {12,12*,1*2,21*,2*1}-avoiding signed permutations in the hyperoctahedral group.
After Wilson's Theorem: if (n+1) is prime then (n+1) is the smallest prime factor of a(n). - Karl-Heinz Hofmann, Aug 21 2024

			G.f. = 2 + 2*x + 3*x^2 + 7*x^3 + 25*x^4 + 121*x^5 + 721*x^6 + 5041*x^7 + ...

C. Stanley Ogilvy and John T. Anderson, Excursions in Number Theory, Oxford University Press, 1966, p. 82.
Wacław Sierpiński, On some unsolved problems of arithmetics, Scripta Mathematica, vol. 25 (1960), p. 125.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

Vincenzo Librandi, Table of n, a(n) for n = 0..300
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 763 and Encyclopedia of Combinatorial Structures 834
T. Mansour and J. West, Avoiding 2-letter signed patterns, arXiv:math/0207204 [math.CO], 2002.
Romeo 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 [math.HO], 2012-2023. - From _N. J. A. Sloane_, Jun 13 2012
Gerard P. Michon, Wilson's Theorem
Hisanori Mishima, Factorizations of many number sequences
Hisanori Mishima, Factorizations of many number sequences
Andrew Walker, Factors of n! +- 1
Arthur T. White, Ringing the changes, Math. Proc. Cambridge Philos. Soc. 94 (1983), no. 2, 203-215.
Robert G. Wilson v, Explicit factorizations
Jun Yan, Results on pattern avoidance in parking functions, arXiv:2404.07958 [math.CO], 2024. See p. 4.
Index entries for sequences related to factorial numbers

Cf. A000142, A002583, A002981, A033312, A051301, A056111, A146968, A217702.

a038507 = (+ 1) . a000142
a038507_list = 2 : f 1 2 where
   f x y = z : f (x + 1) z where z = x * (y - 1) + 1
-- Reinhard Zumkeller, Mar 20 2013

[Factorial(n) +1: n in [0..25]]; // Vincenzo Librandi, Jul 20 2011

Range[0,20]!+1 (* Harvey P. Dale, May 06 2012 *)

A038507(n):= n!+1$
makelist(A038507(n),n,0,30); /* Martin Ettl, Nov 03 2012 */

a(n)=n!+1 \\ Charles R Greathouse IV, Nov 20 2012

from math import factorial
def A038507(n): return factorial(n) + 1 # Karl-Heinz Hofmann, Aug 21 2024

[factorial(n) + 1 for n in range(0,24)] # Stefano Spezia, Apr 21 2025

a(n) = n * (a(n-1) - 1) + 1. - Reinhard Zumkeller, Mar 20 2013
0 = a(n)*(a(n+1) - 5*a(n+2) + 5*a(n+3) - a(n+4)) + a(n+1)*(a(n+1) + a(n+2) - 6*a(n+3) + 2*a(n+4)) + a(n+2)*(3*a(n+2) - a(n+3) - a(n+4)) + a(n+3)*(a(n+3)) if n>=0. - Michael Somos, Apr 23 2014
From Ilya Gutkovskiy, Jan 20 2017: (Start)
E.g.f: exp(x) + 1/(1 - x).
Sum_{n>=0} 1/a(n) = A217702. (End)

Additional comments from Jason Earls, Apr 01 2001
Numericana.com URL fixed by Gerard P. Michon, Mar 30 2010
Entry revised by N. J. A. Sloane, Jun 10 2012

A002583 Largest prime factor of n! + 1.

Original entry on oeis.org

2, 2, 3, 7, 5, 11, 103, 71, 661, 269, 329891, 39916801, 2834329, 75024347, 3790360487, 46271341, 1059511, 1000357, 123610951, 1713311273363831, 117876683047, 2703875815783, 93799610095769647, 148139754736864591, 765041185860961084291, 38681321803817920159601

Offset: 0

N. J. A. Sloane

Theorem: For any N, there is a prime > N. Proof: Consider any prime factor of N!+1.
Cf. Wilson's theorem (1770): p | (p-1)! + 1 iff p is a prime.
If n is in A002981, then a(n) = n!+1. - Chai Wah Wu, Jul 15 2019

			(0!+1)=[2], (1!+1)=[2], (2!+1)=[3], (3!+1)=[7], (4!+1)=25=5*[5], (5!+1)=121=11*[11], (6!+1)=721=7*[103], (7!+1)=5041=71*[71], etc. - Mitch Cervinka (puritan(AT)toast.net), May 11 2009

M. Kraitchik, On the divisibility of factorials, Scripta Math., 14 (1948), 24-26 (but beware errors).
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Georg Fischer, Table of n, a(n) for n = 0..139 (first 101 terms originally derived from Hisanori Mishima's data by T. D. Noe)
A. Borning, Some results for k!+-1 and 2.3.5...p+-1, Math. Comp., 26 (1972), 567-570.
P. Erdős and C. L. Stewart, On the greatest and least prime factors of n! + 1, J. London Math. Soc. (2) 13:3 (1976), pp. 513-519.
M. Kraitchik, On the divisibility of factorials, Scripta Math., 14 (1948), 24-26 (but beware errors). [Annotated scanned copy]
Li Lai, On the largest prime divisor of n! + 1, arXiv:2103.14894 [math.NT], 2021.
Hisanori Mishima, Factorizations of many number sequences
Hisanori Mishima, Factorizations of many number sequences
H. P. Robinson and N. J. A. Sloane, Correspondence, 1971-1972
Blake C. Stacey, Equiangular Lines, Ch. 1, A First Course in the Sporadic SICs, SpringerBriefs in Math. Phys. (2021) Vol. 41, see page 5.
R. G. Wilson v, Explicit factorizations

Cf. A002582, A002981, A038507, A051301, A056111, A096225.

[Maximum(PrimeDivisors(Factorial(n)+1)): n in [0..30]]; // Vincenzo Librandi, Feb 14 2020

PrimeFactors[n_]:=Flatten[Table[ #[[1]],{1}]&/@FactorInteger[n]]; Table[PrimeFactors[n!+1][[ -1]],{n,0,35}] ..and/or.. Table[FactorInteger[n!+1,FactorComplete->True][[ -1,1]],{n,0,35}] (* Vladimir Joseph Stephan Orlovsky, Aug 12 2009 *)
FactorInteger[#][[-1,1]]&/@(Range[0,30]!+1) (* Harvey P. Dale, Sep 04 2017 *)

a(n)=my(f=factor(n!+1)[,1]);f[#f] \\ Charles R Greathouse IV, Dec 05 2012

Erdős & Stewart show that a(n) > n + (1-o(1))log n/log log n and lim sup a(n)/n > 2. - Charles R Greathouse IV, Dec 05 2012

Lai proves that lim sup a(n)/n > 7.238. - Charles R Greathouse IV, Jun 22 2021

More terms from Robert G. Wilson v, Aug 01 2000

Corrected by Jud McCranie, Jan 03 2001

A104366 Smallest prime factor of A104365(n) = A104350(n) + 1.

Original entry on oeis.org

2, 3, 7, 13, 61, 181, 13, 2521, 7561, 103, 415801, 1247401, 167, 191, 211, 127, 23, 40357, 1099944846001, 349, 41, 251, 37, 2243, 146100174169950001, 103, 53, 1217, 1156675078903494150001, 47, 2939, 251, 857, 41, 547, 13127, 47, 48563, 281, 1336484560722851, 479, 373, 2179, 577670972464621571, 17491, 1399, 97, 22893547

Offset: 1

Reinhard Zumkeller, Mar 06 2005

nonn

a(n) = A020639(A104365(n)).

Tyler Busby, Table of n, a(n) for n = 1..170 (terms 1..100 from Chai Wah Wu, terms 101..165 from Max Alekseyev)
Reinhard Zumkeller, Products of largest prime factors of numbers <= n

Cf. A104367, A104358, A051301, A051342, A104358, A104365, A104367, A104368, A104369, A104370, A104371, A104372.

gpf(n) = if (n==1, 1, vecmax(factor(n)[,1])); \\ A006530
spf(n) = if (n==1, 1, vecmin(factor(n)[,1])); \\ A020639
a(n) = spf(prod(i=2, n, gpf(i))+1); \\ Michel Marcus, Feb 21 2023

Corrected by D. S. McNeil, Dec 10 2010

A054415 Smallest prime factor of n!-1 (for n>2), a(2)=1.

Original entry on oeis.org

1, 5, 23, 7, 719, 5039, 23, 11, 29, 13, 479001599, 1733, 87178291199, 17, 3041, 19, 59, 653, 124769, 23, 109, 51871, 625793187653, 149, 20431, 29, 239, 31, 265252859812191058636308479999999, 787, 263130836933693530167218012159999999, 8683317618811886495518194401279999999

Offset: 2

Henry Bottomley, May 10 2000

nonn

The initial term a(2)=1 is not a prime, but it does not affect search results and may prevent submission of duplicates. - M. F. Hasler, Oct 31 2012

			a(3)=5 because 3!-1=5 which is prime; a(5)=7 because 5!-1=119=7*17 and 7<17

Chai Wah Wu, Table of n, a(n) for n = 2..153 (n = 2..135 from Amiram Eldar)
P. Erdős and C. L. Stewart, On the greatest and least prime factors of n! + 1, J. London Math. Soc. (2) 13:3 (1976), pp. 513-519.
M. Kraitchik, On the divisibility of factorials, Scripta Math., 14 (1948), 24-26 (but beware errors). [Annotated scanned copy]
Hisanori Mishima, Factorizations of many number sequences: n! - 1 (n = 1 to 100); Primorials - 1.
R. G. Wilson v, Explicit factorizations

Cf. A002582, A033312, A051301.

Do[ Print[ FactorInteger[ n! - 1, FactorComplete -> True][ [1, 1] ] ], {n, 3, 32} ]

A054415(n)=if(n>2,factor(n!-1)[1,1],1)  \\ M. F. Hasler, Oct 31 2012

Erdős & Stewart show that a(n) > n + (l-o(l))log n/log log n except when n+1 is prime, and that a(n) > n + e(n)sqrt(n) for almost all n where e(n) is any function with lim e(n) = 0. - Charles R Greathouse IV, Dec 05 2012

More terms from Robert G. Wilson v, Aug 01 2000

More terms from Amiram Eldar, Oct 07 2019

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

A051454 a(n) is the smallest prime factor of 1 + lcm(1..k) where k is the n-th prime power A000961(n).

Original entry on oeis.org

2, 3, 7, 13, 61, 421, 29, 2521, 19, 89, 71, 1693, 232792561, 6659, 26771144401, 331, 101, 72201776446801, 1801, 173, 54941, 89, 442720643463713815201, 593, 5171, 239, 1222615931, 103, 7265496855919, 6562349363, 4447, 147099357127, 1931

Offset: 1

Labos Elemer

nonn

			1 + lcm(1..8) = 29^2, so its smallest prime divisor is 29; it occurs as the 7th term in the sequence because 8 is the 7th prime power: A000961(7) = 8.

Sean A. Irvine, Table of n, a(n) for n = 1..82

Cf. A000961, A003418, A049536, A049537, A051301, A051342, A051452.

a:=[]; lcm:=1; for k in [1..83] do if (k eq 1) or IsPrimePower(k) then lcm:=Lcm(lcm,k); a:=a cat [Factorization(1+lcm)[1][1]]; end if; end for; a; // Jon E. Schoenfield, May 28 2018

Join[{2},With[{ppwr=Select[Range[200],PrimePowerQ]},Table[FactorInteger[LCM@@Take[ ppwr,n]+ 1][[1,1]],{n,40}]]] (* Harvey P. Dale, May 28 2024 *)

a(n) = {my(nb = 1, lc = 1, k = 2); while (nb != n, if (isprimepower(k), nb++; lc = lcm(lc, k)); k++;); vecmin(factor(lc +1)[,1]);} \\ Michel Marcus, May 29 2018

from math import prod
from sympy import primepi, integer_nthroot, integer_log, primerange, primefactors
def A051454(n):
    def f(x): return int(n+x-1-sum(primepi(integer_nthroot(x,k)[0]) for k in range(1,x.bit_length())))
    m, k = n, f(n)
    while m != k:
        m, k = k, f(k)
    return min(primefactors(1+prod(p**integer_log(m, p)[0] for p in primerange(m+1)))) # Chai Wah Wu, Aug 15 2024

A096225 a(0) = 1; for n >= 0, a(n+1) = smallest prime factor of a(n)! + 1.

Original entry on oeis.org

1, 2, 3, 7, 71, 6653, 25469, 15750503

Offset: 0

N. J. A. Sloane, Aug 09 2004

			71!+1 is the product of 6653 and a large prime.

Cf. A002583, A051301.

a[1] = 2; a[n_] := Block[{p = PrimePi[a[n - 1]] + 1, r = a[n - 1]! + 1}, While[ Mod[r, Prime[p]] != 0, p++ ]; Prime[p]]; Do[ Print[ a[n]], {n, 7}] (* Robert G. Wilson v, Aug 12 2004 *)
NestList[FactorInteger[#!+1][[1,1]]&,1,7] (* Harvey P. Dale, Sep 20 2016 *)

a(6) and a(7) from Robert G. Wilson v, Aug 12 2004

A362778 Triangular array read by rows: T(n,k) is the least prime factor of n!*k + 1, n >= 1, 1 <= k <= n.

Original entry on oeis.org

2, 3, 5, 7, 13, 19, 5, 7, 73, 97, 11, 241, 19, 13, 601, 7, 11, 2161, 43, 13, 29, 71, 17, 15121, 20161, 11, 30241, 35281, 61, 11, 73, 161281, 449, 241921, 282241, 47, 19, 293, 1088641, 1451521, 23, 2177281, 13, 2903041, 17, 11, 13, 10886401, 233, 18144001, 17, 101, 29030401, 32659201, 43

Offset: 1

Joe B. Stephen, May 03 2023

The primes in each row are distinct because n!*k + 1 are coprime for 1 <= k <= n, and hence this array gives a simple proof that there are infinitely many prime numbers.

			Triangle T(n,k) begins:
  n\k  1    2    3    4    5    6 ...
  1    2
  2    3    5
  3    7   13   19
  4    5    7   73   97
  5   11  241   19   13  601
  6    7   11 2161   43   13   29
  ...

Cf. A362777, A362779.

Cf. A051301 (1st column).

T(n,k) = A020639(A362777(n,k)).

A166862 Primes p that divide n! + 1 for some n other than p-1.

Original entry on oeis.org

2, 7, 11, 19, 23, 29, 43, 47, 59, 61, 67, 71, 79, 83, 103, 109, 127, 131, 137, 139, 149, 163, 179, 191, 193, 199, 227, 233, 239, 251, 257, 263, 269, 271, 277, 293, 307, 311, 317, 347, 359, 367, 379, 383, 389, 397, 401, 419, 431, 443, 449, 461, 463, 467, 479

Offset: 1

Michael B. Porter, Oct 22 2009

nonn

For n >= p, p is one of the factors of n!, so p cannot divide n! + 1. As a result, only 0 <= n <= p-2 needs to be searched.
For n = p-1, by Wilson's Theorem, (p-1)! = -1 (mod p), so p divides (p-1)! + 1.
Since by convention 0! = 1, 2 is included in the sequence as dividing 0!+1 = 2.
The standard heuristic suggests that the fraction of primes in this sequence is 1 - 1/e or about 63%. - Charles R Greathouse IV, Apr 17 2013

			11 is included in the sequence since 11 divides 5! + 1 = 121.
13 is not included in the sequence since the only n for which 13 divides n! + 1 is n = 12.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Wilson's Theorem

Cf. A000142, A038507, A051301, A002583.

isA166862(n) = {local(r);r=0;for(i=0,n-2,if((i!+1)%n==0,r=1));r}

is(p)=my(m=Mod(1,p)); for(k=2,p-2,m*=k; if(m==-1, return(isprime(p)))); p==2 \\ Charles R Greathouse IV, Apr 17 2013

cp's OEIS Frontend

A160245 a(n) = index of the n-th prime in A051301 (least prime factor of m!+1).

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A038507 a(n) = n! + 1.

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A002583 Largest prime factor of n! + 1.

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A104366 Smallest prime factor of A104365(n) = A104350(n) + 1.

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A054415 Smallest prime factor of n!-1 (for n>2), a(2)=1.

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A056111 Highest proper factor of n!+1.

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A051454 a(n) is the smallest prime factor of 1 + lcm(1..k) where k is the n-th prime power A000961(n).