A038507 -id:A038507 - cp's OEIS frontend

A352139 Expansion of e.g.f. 1/(exp(x) - log(1 - x)).

1, -2, 6, -27, 161, -1205, 10799, -113043, 1351461, -18183781, 271784079, -4469044657, 80160267791, -1557710354083, 32597642189657, -730897865864471, 17480390183397209, -444198879957594857, 11951585821669838395, -339434402344422296117

Offset: 0

Seiichi Manyama, Mar 06 2022

sign

Seiichi Manyama, Table of n, a(n) for n = 0..421

Cf. A352138, A352146, A352147.

Cf. A006252, A038507.

m = 19; Range[0, m]! * CoefficientList[Series[1/(Exp[x] - Log[1 - x]), {x, 0, m}], x] (* Amiram Eldar, Mar 06 2022 *)

my(N=20, x='x+O('x^N)); Vec(serlaplace(1/(exp(x)-log(1-x))))

a(n) = if(n==0, 1, -sum(k=1, n, ((k-1)!+1)*binomial(n, k)*a(n-k)));

a(0) = 1; a(n) = -Sum_{k=1..n} ((k-1)! + 1) * binomial(n,k) * a(n-k).

A056110 Highest proper factor of n!-1, or a(n)=1 if n!-1 is not composite.

Original entry on oeis.org

1, 1, 1, 17, 1, 1, 1753, 32989, 125131, 3070523, 1, 3593203, 1, 76922021647, 6880233439, 18720390952421, 108514808571661, 186286524362683, 19499250680671, 2221345311813453913, 10311933282363373211, 498390560021687969, 991459181683, 104102080827724738147651, 19739193437746837432529

Offset: 2

Henry Bottomley, Jun 12 2000

nonn

Amiram Eldar, Table of n, a(n) for n = 2..135
Hisanori Mishima, Factorizations of many number sequences: n! - 1 (n = 1 to 100).
R. G. Wilson v, Explicit factorizations

Cf. A002582.

pf[n_]:=Module[{c=n!-1},If[PrimeQ[c],1,c/FactorInteger[c][[1,1]]]]; Array[pf,30,2] (* Harvey P. Dale, Dec 13 2012 *)

A056110(n)={n=factor(n!-1);if(norml2(n[,2])>1,factorback(n)/n[1,1],1)} \\ M. F. Hasler, Oct 31 2012

a(n) = A033312(n)/A054415(n)

Edited and extended by M. F. Hasler, Oct 31 2012

A068482 Numbers n such that gcd(n!+1,2^n-1)>1.

Original entry on oeis.org

2, 3, 4, 6, 10, 12, 16, 18, 22, 23, 28, 30, 36, 39, 40, 42, 46, 51, 52, 58, 60, 63, 66, 70, 72, 78, 82, 88, 95, 96, 99, 100, 102, 106, 108, 112, 126, 130, 131, 135, 136, 138, 148, 150, 156, 162, 166, 172, 178, 180, 183, 190, 191, 192, 196, 198, 210, 215, 222, 226

Offset: 1

Benoit Cloitre, Mar 10 2002

If n=p-1, p prime, then n is in the sequence.

G. C. Greubel, Table of n, a(n) for n = 1..1000

Cf. A000225 (2^n-1), A038507 (n!+1).

Cf. A068480, A068481, A068483.

Filtered([1..230],n->Gcd(Factorial(n)+1,2^n-1)>1); # Muniru A Asiru, Oct 16 2018

select(n->gcd(factorial(n)+1,2^n-1)>1,[$1..230]); # Muniru A Asiru, Oct 16 2018

Select[Range[300],GCD[#!+1,2^#-1]>1&] (* Harvey P. Dale, Jan 31 2015 *)

isok(n) = gcd(n!+1,2^n-1) > 1; \\ Michel Marcus, Oct 16 2018

A116891 a(n) = gcd(n! + 1, n^n + 1).

Original entry on oeis.org

2, 1, 7, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 47, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 79, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 103, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 127, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 191, 1, 1, 1, 199, 1, 1

Offset: 1

Giovanni Resta, Mar 01 2006

Apparently all the values greater than 1 (cf. A116892) are prime numbers and are equal to 2n+1 with only 4 exceptions for n<82000 (cf. A116894).
From Antti Karttunen, Jul 22 2018: (Start)
The first duplicated value > 1 is 157519 = a(43755) = a(78759). Note that 43755 = 15*2917, while 78759 = 27*2917.
It seems that for the long time after a(1) = 2, all other terms > 1 occur only at such positions k that k+1 is not squarefree. However, this turns out to be false as a(208161) = 555097, and 208162 is a squarefree number.
(End)

			a(3) = gcd(3! + 1, 3^3 + 1) = gcd(7,28) = 7.

Antti Karttunen, Table of n, a(n) for n = 1..80001

Cf. A014566, A038507, A067658, A116892, A116893, A116894.

Table[GCD[n! + 1, n^n + 1], {n, 101}] (* Robert G. Wilson v, Mar 09 2006 *)

A116891(n) = gcd(n!+1,(n^n)+1); \\ Antti Karttunen, Jul 22 2018

A116892 Values of gcd(k!+1, k^k+1), when greater than 1.

Original entry on oeis.org

2, 7, 47, 79, 103, 127, 191, 199, 263, 367, 383, 431, 479, 503, 599, 607, 631, 727, 743, 823, 839, 863, 887, 991, 1087, 1151, 1319, 1367, 1423, 1487, 1511, 1583, 1663, 1783, 1823, 1871, 1951, 2039, 2063, 2111, 2143, 2287, 2311, 2383, 2423, 2447, 2503, 2551

Offset: 1

Giovanni Resta, Mar 01 2006

Apart from the initial term (2) and few exceptional values (A116894) this sequence seems to coincide with A067658. The values of k for which the terms of this sequence are obtained are in A116893.

			gcd(1!+1,1^1+1) = 2 gives the first term;
gcd(3!+1,3^3+1) = gcd(7,28) = 7 gives the second, and so on.

Nick Hobson, Table of n, a(n) for n = 1..10000 (first 1832 terms from Antti Karttunen)

Cf. A014566, A038507, A067658, A116891, A116893, A116894.

C
```
See Links section in A116893.
```

f[n_] := GCD[n! + 1, n^n + 1]; t = Array[f, 1295]; Rest@ Union@ t (* Robert G. Wilson v, Mar 09 2006 *)

lista(nn) = for (n=1, nn, if ((g=gcd(n! + 1, n^n + 1)) != 1, print1(g, ", "))); \\ Michel Marcus, Jul 22 2018

Entries checked by Robert G. Wilson v, Mar 09 2006

A116894 Numbers k such that gcd(k! + 1, k^k + 1) is neither 1 nor 2k+1.

Original entry on oeis.org

1, 5427, 41255, 43755, 208161, 496175, 497135

Offset: 1

Giovanni Resta, Mar 01 2006

g(n) = gcd(n! + 1, n^n + 1) is almost always equal to 1 or to 2n+1. These are the known exceptions: g(1) = 2, g(5427) = 10453, g(41255) = 129341, g(43755) = 157519, g(208161) = 555097. - Hans Havermann, Mar 28 2006

a(8) > 1000000. - Nick Hobson, Feb 20 2024

			gcd(1! + 1, 1^1 + 1) = 2 and 2 != 2*1 + 1, so 1 belongs to the sequence.

Cf. A014566, A038507, A067658, A116891, A116892, A116893.

C
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// See Links section in A116893.
```

a(5) from Hans Havermann, Mar 28 2006

a(6)-a(7) from Nick Hobson, Feb 20 2024

A139149 a(n) = (n!+2)/2.

Original entry on oeis.org

2, 4, 13, 61, 361, 2521, 20161, 181441, 1814401, 19958401, 239500801, 3113510401, 43589145601, 653837184001, 10461394944001, 177843714048001, 3201186852864001, 60822550204416001, 1216451004088320001, 25545471085854720001, 562000363888803840001

Offset: 2

Artur Jasinski, Apr 11 2008

Also the number of (not necessarily maximal) cliques in the (n-1)-(weak) Bruhat graph. - Eric W. Weisstein, Jul 29 2018

			(1!+2)/2 = 3/2 is not an integer.
a(2) = (2!+2)/2 = 2.

Amiram Eldar, Table of n, a(n) for n = 2..450
Eric Weisstein's World of Mathematics, Bruhat Graph.
Eric Weisstein's World of Mathematics, Clique.

a(n) = (n!+m)/m: A038507 (m=1), this sequence (m=2), A139150 (m=3), A139151 (m=4), A139152 (m=5), A139153 (m=6), A139154 (m=7), A139155 (m=8), A139156 (m=9), A139157 (m=10).
Offsets for above sequences are Kempner numbers A002034.
For smallest number of the form (m!+n)/n see A139148.
Cf. A007749, A020458, A082672, A089085, A089130, A117141, A137390, A139056-A139066, A139068, A139070-A139075, A139157, A139159-A139162.

Mathematica
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Table[(n! + 2)/2, {n, 2, 30}]
```

a(n)=n!/2+1 \\ Charles R Greathouse IV, Jun 20 2012

A301593 n can be represented the sum of a(n) distinct factorials. (If there is no such representation, a(n) = 0.)

Original entry on oeis.org

1, 1, 2, 0, 0, 1, 2, 2, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 2, 3, 0, 0, 2, 3, 3, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 1

Seiichi Manyama, Mar 24 2018

nonn

			n |                        | a(n)
--+------------------------+-----
1 | 1!                     |  1
2 | 2!                     |  1
3 | 1! + 2!                |  2
6 | 3!                     |  1
7 | 3! + 1!                |  2
8 | 3! + 2!                |  2
9 | 3! + 2! + 1!           |  3

Seiichi Manyama, Table of n, a(n) for n = 1..10000
Wikipedia, Factorial number system

Cf. A000142, A007489, A038507, A059590, A108731, A115944, A301523.

a(n!) = 1, a(n!+1) = 2.

A354893 a(n) = n! * Sum_{d|n} d^(n - d) / (n/d)!.

Original entry on oeis.org

1, 3, 7, 73, 121, 12361, 5041, 5308801, 44452801, 5681370241, 39916801, 16800125569921, 6227020801, 35897693762810881, 2134168822456070401, 190139202281277849601, 355687428096001, 3563095308471181273190401, 121645100408832001

Offset: 1

Seiichi Manyama, Jun 10 2022

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..291

Cf. A038507, A342628, A354845, A354891, A354892.

a[n_] := n! * DivisorSum[n, #^(n - #)/(n/#)! &]; Array[a, 19] (* Amiram Eldar, Jun 10 2022 *)

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

my(N=20, x='x+O('x^N)); Vec(serlaplace(sum(k=1, N, (exp((k*x)^k)-1)/k^k)))

E.g.f.: Sum_{k>0} (exp((k * x)^k) - 1)/k^k.

If p is prime, a(p) = 1 + p! = A038507(p).

A036739 a(n) = (n!)^n+1.

Original entry on oeis.org

2, 2, 5, 217, 331777, 24883200001, 139314069504000001, 82606411253903523840000001, 6984964247141514123629140377600000001, 109110688415571316480344899355894085582848000000001, 395940866122425193243875570782668457763038822400000000000000000001

Offset: 0

G. L. Honaker, Jr.

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..30

Cf. A019514, A020549, A036740, A038507.

List([0..11],n->Factorial(n)^n+1); # Muniru A Asiru, Mar 19 2018

seq(factorial(n)^n+1,n=0..11); # Muniru A Asiru, Mar 19 2018

Table[(n!)^n+1,{n,0,10}] (* Harvey P. Dale, Apr 10 2012 *)

a(n) = (n!)^n + 1; \\ Altug Alkan, Mar 19 2018

a(n) ~ (2*Pi)^(n/2) * n^(n^2 + n/2) / exp(n^2 - 1/12). - Vaclav Kotesovec, Mar 19 2018

One more term from Harvey P. Dale, Apr 10 2012

cp's OEIS Frontend

A352139 Expansion of e.g.f. 1/(exp(x) - log(1 - x)).

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A056110 Highest proper factor of n!-1, or a(n)=1 if n!-1 is not composite.

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A068482 Numbers n such that gcd(n!+1,2^n-1)>1.

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A116891 a(n) = gcd(n! + 1, n^n + 1).

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A116892 Values of gcd(k!+1, k^k+1), when greater than 1.

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C

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A116894 Numbers k such that gcd(k! + 1, k^k + 1) is neither 1 nor 2k+1.

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C

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A139149 a(n) = (n!+2)/2.

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