A038507 -id:A038507 - cp's OEIS frontend

A078145 Length of period of continued fraction for square root of 1+n!.

1, 2, 4, 0, 0, 40, 0, 153, 558, 3074, 4285, 22602, 180544, 766146, 766082, 524570, 9932193, 5193268, 763601450, 4284694240, 3059999982, 48742214702, 6354126960, 1975806716944, 929707144775

Offset: 1

Labos Elemer, Nov 25 2002

			Period for sqrt(3!+1) = sqrt(7) = {1,1,1,4}, a(3) = 4.

Cf. A003285, A038507, A078146, A064025.

Table[Length[Last[ContinuedFraction[Sqrt[1+n! ]]]], {n, 1, 14}]

a(n) = A003285(A038507(n)). - Michel Marcus, Sep 25 2019

a(15)-a(20) from Vaclav Kotesovec, Aug 28 2019
a(21) from Chai Wah Wu, Sep 23 2019
a(22)-a(23) from Chai Wah Wu, Sep 25 2019
a(24)-a(25) from Chai Wah Wu, Oct 01 2019

A082995 Distance from n!+1 to next larger square.

Original entry on oeis.org

2, 1, 2, 0, 0, 8, 0, 80, 728, 224, 323, 39168, 82943, 176399, 215295, 3444735, 26167683, 114349224, 255004928, 1158920360, 11638526760, 42128246888, 191052974115, 97216010328, 2430400258224, 1553580508515, 4666092737475

Offset: 1

Jason Earls, May 29 2003

nonn

The only known values of n such that n!+1 is a perfect square are 4, 5 and 7. Paul Leyland, et al. have found no other solutions for n <= 1 million (see link). For 1 <= n <= 11, n!+1 is within 1000 of being a square. Is there another n such that n!+1 <= "1000 away" from being a perfect square?

			a(5)=0 because 5!+1 is a square.
a(8)=80 because 8!+1 = 40321 and the next larger square is 40401, so 40401-40321 = 80.

Amiram Eldar, Table of n, a(n) for n = 1..808
P. Leyland, Solutions to n!+1=m^2.

Cf. A038507, A087374.

a[n_] := Ceiling @ Sqrt[(f = n! + 1)]^2 - f; Array[a, 27] (* Amiram Eldar, Dec 14 2019 *)

for(k=1,27,print1(ceil(sqrt(k!+1))^2-(k!+1),", ")) \\ Hugo Pfoertner, Dec 14 2019

A137244 a(n) = lcm_{k=0..n} (k! + 1).

Original entry on oeis.org

2, 2, 6, 42, 1050, 127050, 13086150, 65967282150, 2659866783570150, 13594579130827036650, 4484729304047661947505150, 179016047168539016473835519025150, 85748973198421705721932588223712809265150, 533960639770963461900374948788827304744234574385150

Offset: 0

Karl Levy, Mar 09 2008

I came upon this sequence in an attempt to solve an open Erdős problem: Show that Sum_{k>=0} 1/(k!+1) is rational/irrational/transcendental.

Harvey P. Dale, Table of n, a(n) for n = 0..43

Cf. A038507, A000142.

With[{t=Range[0,20]!+1},Table[LCM@@Take[t,n],{n,Length[t]}]] (* Harvey P. Dale, Dec 21 2015 *)

a(n) = {lc = 1; for (k=0, n, lc = lcm(lc, k!+1);); return (lc);} \\ Michel Marcus, Jul 25 2013

a(n) = lcm_{k=0..n} (k! + 1).

More terms from Harvey P. Dale, Dec 21 2015

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

A185387 Expansion of e.g.f. exp(x)+log(1/(1-x)).

Original entry on oeis.org

1, 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

Vladimir Kruchinin, Feb 21 2011

nonn

Apart from the offset, the same as A038507. - R. J. Mathar, Jun 07 2011

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

Cf. A038507.

[1] cat [(1+Factorial(n-1)): n in [1..20]]; // Vincenzo Librandi, Oct 04 2011

Join[{1},Range[0,20]!+1] (* Harvey P. Dale, Jun 28 2011 *)

Vec(serlaplace(exp(x)+log(1/(1-x)))) \\ Joerg Arndt, Mar 03 2011

a(n) = (n-1)!+1, n>0, a(0)=1.
From Sergei N. Gladkovskii, Jun 14 2012: (Start)
G.f.: 1 + x*G(0) where G(k) = 1 + k!/(1 - x/(x + (k!)/G(k+1))); (continued fraction).
E.g.f.: 1 + x*G(0) where G(k) = 1 + k!/(1 - x/(x + (k+2)*(k!)/G(k+1))); (continued fraction).
(End).

A217702 Decimal expansion of Sum_{n>=0} 1/(1+n!).

Original entry on oeis.org

1, 5, 2, 6, 0, 6, 8, 1, 3, 4, 4, 7, 3, 3, 3, 0, 8, 2, 4, 7, 7, 8, 0, 4, 7, 2, 2, 5, 1, 6, 2, 4, 3, 8, 4, 0, 5, 4, 4, 9, 7, 5, 4, 2, 4, 0, 4, 6, 6, 4, 6, 5, 6, 5, 0, 7, 5, 8, 3, 6, 5, 6, 5, 1, 6, 7, 9, 5, 7, 4, 7, 1, 1, 4, 7, 2, 0, 2, 5, 2, 3, 8, 8, 0, 1, 8, 1, 6, 0, 9, 7, 3, 1, 1, 7, 7, 3, 6, 3, 0

Offset: 1

Jean-François Alcover, Mar 20 2013

Is there any known closed form for this sum?

			1.526068134473330824778047225162438405449754240466465650758365651679...

Cf. A038507.

digits = 100; NSum[1/(1 + n!), {n, 0, Infinity}, NSumTerms -> digits, WorkingPrecision -> digits+5] // RealDigits[#, 10, digits]& // First

A229554 a(n) = 7*n! + 1.

Original entry on oeis.org

8, 8, 15, 43, 169, 841, 5041, 35281, 282241, 2540161, 25401601, 279417601, 3353011201, 43589145601, 610248038401, 9153720576001, 146459529216001, 2489811996672001, 44816615940096001, 851515702861824001, 17030314057236480001, 357636595201966080001

Offset: 0

Vincenzo Librandi, Sep 30 2013

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

Cf. sequences of the type k*n!+1: A038507 (k=1), A052898 (k=2), A173324 (k=3), A173322 (k=4), A173319 (k=5), A173314 (k=6), this sequence (k=7).

Magma
```
[7*Factorial(n)+1: n in [0..25]];
```

[8] cat [n eq 1 select n+7 else n*Self(n-1)-n+1: n in [1..25]];

Mathematica
```
Table[7 n! + 1, {n, 0, 25}]
```

a(0)=8, a(n) = n*a(n-1)-n+1.

E.g.f.: 7/(1-x)+exp(x). - Geoffrey Critzer, Sep 30 2013

A230459 Ordered by increasing m with k < m, a(n) is the n-th record value of gcd(k!+1, m!+1).

Original entry on oeis.org

2, 7, 71, 661, 733, 2371, 3529, 13499, 46549, 98101, 163517, 197933, 1924217, 3322441, 5370731

Offset: 1

James G. Merickel, Oct 19 2013

The pairs (m,k) generating records are (1,0), (6,3), (9,7), (17,8), (89,51), (174,144), (349,228), (422,81), (650,406), (1415,1718), (1697,161), (1622,773), (1884,1219), (7003,2031) and (17057,660).
Heuristics in concert with a database of 'small' (less than, say, 10^12) prime factors of numbers of this kind would generate faster accurate results with near certainty, while any truly proving program is doomed to be relatively slow by comparison (and see following on a(15)).
Note: An auxiliary program employed a limit of 10^8--in lieu of a database--to generate the likely-but-not-certain value of a(15) shown last.

			a(1)=2, corresponding to m=1 and k=0.  7 is the first value other than 1 to be the greatest common divisor of two different numbers k!+1 and m!+1, where m is increasing and k is allowed to increase to m-1 for a given m (for m=6 and k=3, m!+1=7*103 and k!+1=7); so that a(2)=7.

Cf. A002583, A002981, A038507, A051301.

{
\\ The constant L here is arbitrary.\\
\\ This does not generate a(1).\\
rec=2;L=10000;F=vector(L);n=2;
for(k=1,L,n--;n*=k;n++;F[k]=n);
for(m=2,L,
  for(k=1,m-1,
    a=gcd(F[m],F[k]);if(a>rec,
      rec=a;print1(a": "m","k"\n"))))
}

A277080 Irregular triangle read by rows: T(n,k) = number of size k subsets of S_n that remain unchanged by reverse.

Original entry on oeis.org

1, 1, 1, 1, 1, 0, 1, 1, 0, 3, 0, 3, 0, 1, 1, 0, 12, 0, 66, 0, 220, 0, 495, 0, 792, 0, 924, 0, 792, 0, 495, 0, 220, 0, 66, 0, 12, 0, 1, 1, 0, 60, 0, 1770, 0, 34220, 0, 487635, 0, 5461512, 0, 50063860, 0, 386206920, 0, 2558620845, 0, 14783142660, 0, 75394027566

Offset: 0

Christian Bean, Sep 28 2016

The reverse of a permutation is the reverse in one line notation. For example the reverse of 43521 is 12534.

T(n,k) is the number of size k bases of S_n which remain unchanged by reverse.

			For n = 4 and k = 2 the subsets that remain unchanged by reverse are {4321, 1234}, {1243, 3421}, {4231, 1324}, {1342, 2431}, {1423, 3241}, {1432, 2341}, {2134, 4312}, {3412, 2143}, {2314, 4132}, {3142, 2413}, {4213, 3124} and {4123, 3214} so T(4,2) = 12.
For n = 3 and k = 4 the subsets that remain unchanged by reverse are {231, 321, 132, 123}, {321, 213, 312, 123} and {231, 132, 312, 213} so T(3,4) = 3.
The triangle starts:
1, 1;
1, 1;
1, 0, 1;
1, 0, 3, 0, 3, 0, 1;

Row lengths give A038507.

def T(n,k):
    if k % 2 == 1:
        return 0
    return binomial( factorial(n)/2, k/2 )

T(n,k) = C(n!/2, k/2) if k is even and T(n,k) = 0 if k is odd.

A277083 Irregular triangle read by rows: T(n,k) = number of size k subsets of S_n that remain unchanged by a rotation of 180 degrees.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 2, 3, 4, 3, 2, 1, 1, 8, 36, 120, 322, 728, 1428, 2472, 3823, 5328, 6728, 7728, 8092, 7728, 6728, 5328, 3823, 2472, 1428, 728, 322, 120, 36, 8, 1, 1, 8, 84, 504, 3178, 15512, 74788, 311144, 1252819, 4577328, 16087512, 52691408, 165911284

Offset: 0

Christian Bean, Sep 28 2016

A permutation, p, can be thought of as a set of points (i, p(i)). If you plot all the points and rotate the picture by 180 degrees then you get a permutation back.

T(n,k) is the number of size k subsets of S_n that remain unchanged by a rotation of 180 degrees.

			For n = 3 and k = 3, the subsets unchanged by rotating 180 degrees are {213,132,123}, {231,312,123}, {321,132,213} and {321,231,312} so T(3,3) = 4.
Triangle starts:
1, 1;
1, 1;
1, 2, 1;
1, 2, 3, 4, 3, 2, 1;

Row lengths give A038507.

Cf. A037223.

T(n,k) = Sum_( binomial( n! - R(n), i ) * binomial( R(n), k-2*i ) for i in [0..floor(k/2)] ) where R(n) = A037223(n).

cp's OEIS Frontend

A078145 Length of period of continued fraction for square root of 1+n!.

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A082995 Distance from n!+1 to next larger square.

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PARI

A137244 a(n) = lcm_{k=0..n} (k! + 1).

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

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A185387 Expansion of e.g.f. exp(x)+log(1/(1-x)).

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Magma

Mathematica

PARI

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A217702 Decimal expansion of Sum_{n>=0} 1/(1+n!).

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Mathematica

A229554 a(n) = 7*n! + 1.

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Magma

Magma

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A230459 Ordered by increasing m with k < m, a(n) is the n-th record value of gcd(k!+1, m!+1).

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