A039716 -id:A039716 - cp's OEIS frontend

A177771 a(n) = (prime(n) - 1)!.

1, 2, 24, 720, 3628800, 479001600, 20922789888000, 6402373705728000, 1124000727777607680000, 304888344611713860501504000000, 265252859812191058636308480000000

Offset: 1

Giovanni Teofilatto, May 13 2010

nonn

By Wilson's theorem, a(n) = -1 (mod p) where p is the n-th prime. - Charles R Greathouse IV, Sep 04 2013

Paulo Ribenboim, The Little Book of Bigger Primes, Springer-Verlag NY 2004. See p. 21.

Vincenzo Librandi, Table of n, a(n) for n = 1..80

Cf. A006093, A039716.

[Factorial(p-1): p in PrimesUpTo(20)]; // Vincenzo Librandi, May 31 2013

(Prime[Range[12]]-1)! (* Harvey P. Dale, Jul 12 2011 *)

apply(p->(p-1)!,primes(10)) \\ Charles R Greathouse IV, Sep 04 2013

a(n) = A010050(A005097(n-1)). a(n)^2 = A177926(n). - R. J. Mathar, May 24 2010

A080087 Number of factors of 5 in the factorial of the n-th prime, counted with multiplicity.

Original entry on oeis.org

0, 0, 1, 1, 2, 2, 3, 3, 4, 6, 7, 8, 9, 9, 10, 12, 13, 14, 15, 16, 16, 18, 19, 20, 22, 24, 24, 25, 25, 26, 31, 32, 33, 33, 35, 37, 38, 39, 40, 41, 43, 44, 46, 46, 47, 47, 51, 53, 55, 55, 56, 57, 58, 62, 63, 64, 65, 66, 68, 69, 69, 71, 75, 76, 76, 77, 81, 82, 84, 84, 86, 87, 89, 90

Offset: 1

Paul D. Hanna, Jan 26 2003

nonn

Highest power of 5 dividing prime(n)! = A039716(n), or also the number of trailing end 0's in A039716(n). - Lekraj Beedassy, Oct 31 2010

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A080084, A080085, A080086, A039716, A112765, A027868.

R:= NULL: v:= 0: p:= 0:
for i from 1 to 100 do
   q:= p;
   p:= nextprime(p);
   v:= v + add(1+padic:-ordp(x,5), x = 1+floor(q/5) .. floor(p/5));
   R:= R,v;
od:
R; # Robert Israel, Sep 27 2023

lst={};Do[p=Prime[n];s=0;While[p>1,p=IntegerPart[p/5];s+=p;];AppendTo[lst,s],{n,5!}];lst (* Vladimir Joseph Stephan Orlovsky, Jul 28 2009 *)

a(n) = valuation(prime(n)!, 5); \\ Michel Marcus, Jan 15 2015

a(n) = Sum_{k=1..L} floor(prime(n)/5^k), where L = log(p_n)/log(5).

a(n) = A112765(A039716(n)). - Michel Marcus, Sep 28 2023

A100124 Decimal expansion of Sum_{n>0} 1/prime(n)!.

Original entry on oeis.org

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

Offset: 0

Mark Hudson (mrmarkhudson(AT)hotmail.com), Nov 11 2004

Mingarelli shows that this constant is irrational. - Charles R Greathouse IV, Nov 05 2013
Convergence follows because A100124 < e - 2 = 0.71828... = 1/2! + 1/3! + 1/4! + 1/5! because e - 2 contains every term in A100124. The relation to e suggests a different question: is this constant not just irrational but also transcendental? - Timothy Varghese, May 07 2014
This is e times the probability that a prime is chosen from a Poisson distribution with lambda = 1. - Charles R Greathouse IV, Dec 07 2014

			0.67519843791111434190056160759135729953927678856513265156034106451688586148...

Angelo B. Mingarelli, Abstract factorials, arXiv:0705.4299 [math.NT], 2007-2012.

Cf. A000720, A010051, A039716, A094289.

RealDigits[Sum[1/Prime[n]!, {n, 1, 20}], 10, 100][[1]] (* Amiram Eldar, Nov 25 2020 *)

default(realprecision,100); sum(n=1,100,1/(prime(n)!),0.)

prec=exp(lambertw(default(realprecision)/exp(1)*log(10))+1)+9; P=s=.5;p=2;forprime(q=3,prec,P/=prod(i=p+1,q,i);s+=P;p=q); s \\ Charles R Greathouse IV, Nov 05 2013

Equals Sum_{k>0} A010051(k)/k!. - R. J. Cano, Jan 25 2017
From Amiram Eldar, Nov 25 2020: (Start)
Equals Sum_{k>=1} 1/A039716(k).
Equals Sum_{k>=1} pi(k)/((k+1)*(k-1)!), where pi = A000720. (End)

A177946 a(n) = prime(n)! / n!.

Original entry on oeis.org

2, 3, 20, 210, 332640, 8648640, 70572902400, 3016991577600, 71241227785728000, 2436552577639909048320000, 205999445200465037721600000, 28734252852655074735274328064000000, 5372155913332392772506888374845440000000

Offset: 1

Reinhard Zumkeller, May 15 2010

nonn

Index entries for sequences related to factorial numbers.

Cf. A000142, A039716, A157132.

a(n) = prime(n)! / n!; \\ Michel Marcus, Sep 08 2015

a(n) = A039716(n)/A000142(n).

More terms from Michel Marcus, Sep 08 2015

A261997 a(n) = prime(n)! - prime(n!).

Original entry on oeis.org

0, 3, 107, 4951, 39916141, 6227015357, 355687428046967, 121645100408347963, 25852016738884971417571, 8841761993739701954543554805353, 8222838654177922817725562105174617

Offset: 1

Altug Alkan, Sep 08 2015

nonn

a(n) is prime for n = 2, 3, 4, 5, 7.

			For n=2, a(n) = prime(n)! - prime(n!) = prime(2)! - prime(2!) = 3.

Charles R Greathouse IV, Table of n, a(n) for n = 1..21

Cf. A039716, A062439.

Array[Prime[#]! - Prime[#!] &, {11}] (* Michael De Vlieger, Sep 08 2015 *)

PARI
```
vector(11, n, prime(n)! - prime(n!))
```

a(n) = A039716(n) - A062439(n).

log a(n) ~ n log^2 n. - Charles R Greathouse IV, Sep 08 2015

A305618 Expansion of e.g.f. log(1 + Sum_{k>=1} x^prime(k)/prime(k)!).

Original entry on oeis.org

0, 1, 1, -3, -9, 20, 190, -126, -6280, -10326, 293041, 1519320, -16985045, -194560444, 1013712777, 27317463952, -19210030599, -4305097718760, -17733269020226, 743855089334604, 7868686621862292, -132351392654695270, -2854492900112993039, 20150897206881256464

Offset: 1

Ilya Gutkovskiy, Jun 06 2018

sign

Logarithmic transform of A010051.

			E.g.f.: A(x) = x^2/2! + x^3/3! - 3*x^4/4! - 9*x^5/5! + 20*x^6/6! + ...
exp(A(x)) = 1 + x^2/2! + x^3/3! + x^5/5! + x^7/7! + ... + x^A000040(k)/A039716(k) + ...
exp(exp(A(x))-1) = 1 + x^2/2! + x^3/3! + 3*x^4/4! + 11*x^5/5! + ... + A190476(k)*x^k/k! + ...

Alois P. Heinz, Table of n, a(n) for n = 1..472

Cf. A000040, A002120, A007447, A010051, A039716, A190476, A218002, A303073, A305619.

a:= proc(n) option remember; (t-> `if`(n=0, 0, t(n)-add(a(j)*t(n-j)*
       j*binomial(n, j), j=1..n-1)/n))(i-> `if`(isprime(i), 1, 0))
    end:
seq(a(n), n=1..25);  # Alois P. Heinz, Dec 04 2018

nmax = 24; Rest[CoefficientList[Series[Log[1 + Sum[x^Prime[k]/Prime[k]!, {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!]
a[n_] := a[n] = Boole[PrimeQ[n]] - Sum[k Binomial[n, k] Boole[PrimeQ[n - k]] a[k], {k, 1, n - 1}]/n; a[0] = 0; Table[a[n], {n, 24}]

A055929 Euler totient function of the factorial of prime(n).

Original entry on oeis.org

1, 2, 32, 1152, 8294400, 1194393600, 64210599936000, 20804234379264000, 4229084764616785920000, 1396531754239566739931136000000, 1256878578815610065938022400000000, 2046959290878571310305421983481856000000000, 4853749870531268290996216607232176947200000000000

Offset: 1

Labos Elemer, Jul 17 2000

nonn

Amiram Eldar, Table of n, a(n) for n = 1..87

Cf. A000010, A000142, A039716, A048855.

a[n_] := EulerPhi[Prime[n]!]; Array[a, 12] (* Amiram Eldar, Jun 03 2024 *)

a(n) = eulerphi(prime(n)!); \\ Amiram Eldar, Jun 03 2024

a(n) = phi(prime(n)!) = A000010(A039716(n)) = A048855(prime(n)).

a(12)-a(13) from Amiram Eldar, Jun 03 2024

A055930 Number of distinct prime factors of totient of (n-th prime)!.

Original entry on oeis.org

0, 1, 1, 2, 3, 3, 4, 4, 5, 6, 6, 7, 8, 8, 9, 9, 10, 10, 11, 11, 11, 12, 13, 14, 15, 15, 15, 16, 16, 16, 18, 18, 19, 19, 21, 21, 21, 22, 23, 23, 24, 24, 24, 24, 25, 25, 27, 29, 30, 30, 30, 30, 30, 30, 31, 32, 32, 32, 33, 34, 34, 34, 36, 36, 36, 37, 38, 39, 40, 40, 40, 41, 42, 42

Offset: 1

Labos Elemer, Jul 17 2000

nonn

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

Cf. A001221, A039716.

Table[PrimeNu[EulerPhi[Prime[n]!]], {n, 1, 50}] (* G. C. Greubel, May 13 2017 *)

for(n=1,50, print1(omega(eulerphi(prime(n)!)) , ", ")) \\ G. C. Greubel, May 13 2017

a(n) = A001221(A055929(n)).

A114574 a(n) = p*p! where p = prime(n).

Original entry on oeis.org

4, 18, 600, 35280, 439084800, 80951270400, 6046686277632000, 2311256907767808000, 594596384994354462720000, 256411097818451356681764864000000, 254907998279515607349492449280000000, 509258864375374766713691244518493388800000000, 1371553591139716091434972544191070818271232000000000

Offset: 1

L. Shawnee Cook (shawneecook(AT)gmail.com), Feb 17 2006

nonn

			2 * 2! = 4, 3 * 3! = 18, 5 * 5! = 600.

Originally published on Nov 09 2004 by indiejade(AT)gmail.com

Cf. A000040 (prime(n)), A039716 (prime(n)!).

[p*Factorial(p): p in PrimesUpTo(50)]; // Vincenzo Librandi, Jun 09 2013

Array[Prime[#] Prime[#]! &, 13] (* Michael De Vlieger, Aug 04 2015 *)
#*#!&/@Prime[Range[15]] (* Harvey P. Dale, Sep 21 2024 *)

from sympy import factorial, prime
def a(n): p = prime(n); return p * factorial(p)
print([a(n) for n in range(1, 14)]) # Michael S. Branicky, Apr 18 2021

a(n) = A000040(n)*A039716(n). - Michel Marcus, Feb 08 2015

a(n) = A001563(A000040(n)). - Michel Marcus, Aug 04 2015

A131492 Numbers n such that the sum of the Carmichael lambda functions of the divisors is a proper divisor of n.

Original entry on oeis.org

140, 189, 378, 1375, 2750, 2775, 2997, 4524, 5550, 5661, 5994, 6375, 11253, 11322, 12750, 13416, 13505, 22506, 25925, 27010, 27511, 30613, 32208, 32513, 32760, 45917, 49665, 49959, 51850, 55022, 61061, 61226, 65026, 67488, 91834, 93605

Offset: 1

R. J. Mathar, Jul 29 2007

nonn

The auxiliary sequence defined by b(n)=sum_{d|n} A002322(d) starts 1,2,3,4,5,6,7,6,9,10,11,10,13,14,11,10,17,18,19,16,...

The auxiliary sequence is A141258. [Reinhard Zumkeller, Feb 17 2012]

Amiram Eldar, Table of n, a(n) for n = 1..1000
W. D. Banks and F. Luca, On integers with a special divisibility property, Archivum Mathematicum (BRNO) 42 (2006) pp 31-42.

Cf. A002322, A039716.

Select[ Range[100000], Divisible[#, s = Total[ CarmichaelLambda /@ Divisors[#]]] && s < # &] (* Jean-François Alcover, Jun 24 2013 *)

lambda(p,alpha)={ if(p>=3 || alpha<=2, return(p^(alpha-1)*(p-1)), return(2^(alpha-2)) ; ) ; } A002322(n)={ local(pf,rmax,resul) ; if(n==1, return(1) ) ; pf=factor(n) ; rmax=matsize(pf)[1] ; resul= lambda(pf[1,1],pf[1,2]) ; for(r=2,rmax, resul=lcm(resul,lambda(pf[r,1],pf[r,2])) ; ) ; return(resul) ; } b(n)={ sumdiv(n,d,A002322(d)) ; } { for(n=1,120000, l=b(n) ; if( l != 1 && l != n && n%l==0, print1(n,",") ) ; ) ; }

n such that (sum_{d|n} A002322(d)) | n.

cp's OEIS Frontend

A177771 a(n) = (prime(n) - 1)!.

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A080087 Number of factors of 5 in the factorial of the n-th prime, counted with multiplicity.

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A100124 Decimal expansion of Sum_{n>0} 1/prime(n)!.

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A177946 a(n) = prime(n)! / n!.

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A261997 a(n) = prime(n)! - prime(n!).

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A305618 Expansion of e.g.f. log(1 + Sum_{k>=1} x^prime(k)/prime(k)!).

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Maple

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A055929 Euler totient function of the factorial of prime(n).

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