Displaying 1-10 of 11 results found.
Smallest prime factor of A020549(n) = (n!)^2 + 1.
+20
6
2, 2, 5, 37, 577, 14401, 13, 101, 17, 131681894401, 13168189440001, 1593350922240001, 101, 38775788043632640001, 29, 1344169, 149, 9049, 37, 710341, 41, 61, 337, 509, 384956219213331276939737002152967117209600000001, 941
COMMENTS
By construction, for n >= 2, a(n) == 1 (mod 4) and a(n) > n.
The first member of A104636 for which a(n) < 2*n+1 is 48.
a(a(n)-n-1) = a(n). (End)
REFERENCES
T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 147.
MAPLE
f:= proc(n) local a;
a:= min(map(proc(t) if t[1]::integer then t[1] fi end proc, ifactors((n!)^2+1, easy)[2]));
if a = infinity then
a:= traperror(timelimit(60, min(map(t -> t[1], ifactors((n!)^2+1)[2]))));
fi;
a
end proc:
MATHEMATICA
Join[{2}, Array[FactorInteger[(#!)^2 + 1][[1, 1]]&, {25}]] (* Vincenzo Librandi, Feb 28 2017 *)
PROG
(Magma) [2] cat [Min(PrimeFactors(Factorial(n)^2 + 1)):n in[1..25]]; // Vincenzo Librandi, Feb 28 2017
Numbers n such that A020549(n)=(n!)^2+1 is a semiprime.
+20
2
6, 7, 8, 12, 15, 16, 17, 18, 19, 28, 29, 41, 45, 53, 55, 61, 73
COMMENTS
The smaller of the two prime factors is given in A083341. The next candidates for a continuation are 55 and 61. (55!)^2 + 1 is composite with 147 decimal digits and unknown factorization.
(55!)^2 + 1 has been factored using ECM into P52*P96 with P52 = A083341(15). (61!)^2 + 1 is composite with 168 decimal digits. - Hugo Pfoertner, Jul 13 2019
Using CADO-NFS, (61!)^2 + 1 has been factored into P58*P110 with P58 = A282706(61) in 17 days wall clock time using 56 million CPU seconds. a(18) >= 75. - Hugo Pfoertner, Aug 04 2019
LINKS
The CADO-NFS Development Team, Cado-NFS, An Implementation of the Number Field Sieve Algorithm, Release 2.3.0, 2017
EXAMPLE
a(1)=6 because (6!)^2+1=518401=13*39877 is a semiprime.
MATHEMATICA
Select[Range[60], PrimeOmega[(#!)^2+1]==2&] (* Harvey P. Dale, Dec 12 2018 *)
Largest prime factor of A020549(n) = (n!)^2 + 1.
+20
1
2, 2, 5, 37, 577, 14401, 39877, 251501, 95629553, 131681894401, 13168189440001, 1593350922240001, 2271708245569901, 38775788043632640001, 2404319663572286441, 1272170577304043929, 2938007628841577533852349, 13980942259426143240713449, 1107848353183710355135404972973, 20831587158104092560535861261
MAPLE
a:= n-> max(numtheory[factorset](n!^2+1)):
PROG
(PARI) a(n) = vecmax(factor(n!^2 + 1)[, 1]); \\ Daniel Suteu, Jun 10 2022
a(n) = (n!)^2.
(Formerly M3666 N1492)
+10
122
1, 1, 4, 36, 576, 14400, 518400, 25401600, 1625702400, 131681894400, 13168189440000, 1593350922240000, 229442532802560000, 38775788043632640000, 7600054456551997440000, 1710012252724199424000000, 437763136697395052544000000, 126513546505547170185216000000
COMMENTS
Let M_n be the symmetrical n X n matrix M_n(i,j) = 1/Max(i,j); then for n > 0 det(M_n)=1/a(n). - Benoit Cloitre, Apr 27 2002
The n-th entry of the sequence is the value of the permanent of a k X k matrix A defined as follows: k is the n-th odd number; if we concatenate the rows of A to form a vector v of length n^2, v_{i}=1 if i=1 or a multiple of 2. - Simone Severini, Feb 15 2006
a(n) = number of set partitions of {1,2,...,3n-1,3n} into blocks of size 3 in which the entries of each block mod 3 are distinct. For example, a(2) = 4 counts 123-456, 156-234, 126-345, 135-246. - David Callan, Mar 30 2007
Number of permutations of {1,2,...,2n} with no even entry followed by a smaller entry. Example: a(2)=4 because we have 1234, 1324, 3124 and 2314.
Number of permutations of {1,2,...,2n} with n even entries that are followed by a smaller entry. Example: a(2)=4 because we have 2143, 3421, 4213 and 4321.
Number of permutations of {1,2,...,2n-1} with no even entry followed by a smaller entry. Example: a(2)=4 because we have 123, 132, 312 and 231.
Number of permutations of {1,2,...,2n-1} with n-1 odd entries followed by a smaller entry. Example: a(2)=4 because we have 132, 312, 231 and 321.
(End)
G. Leibniz in his "Ars Combinatoria" established the identity P(n)^2 = P(n-1)[P(n+1)-P(n)], where P(n) = n!. (For example, see the Burton reference.) - Mohammad K. Azarian, Mar 28 2008
a(n) is also the determinant of the symmetric n X n matrix M defined by M(i,j) = sigma_2(gcd(i,j)) for 1 <= i,j <= n, and n>0, where sigma_2 is A001157. - Enrique Pérez Herrero, Aug 13 2011
The o.g.f. of 1/a(n) is BesselI(0,2*sqrt(x)). See Abramowitz-Stegun (reference and link under A008277), p. 375, 9.6.10. - Wolfdieter Lang, Jan 09 2012
Number of n x n x n cubes C of zeros and ones such that C(x,y,z) and C(u,v,w) can be nonzero simultaneously only if either x!=u, y!=v, or z!=w. This generalizes permutations which can be considered as n x n squares P of zeros and ones such that P(x,y) and P(u,v) can be nonzero simultaneously only if either x!=u or y!=v. - Joerg Arndt, May 28 2012
a(n) is the number of functions f:[n]->[n(n+1)/2] such that, if round(sqrt(2f(x))) = round(sqrt(2f(y))), then x=y. - Dennis P. Walsh, Nov 26 2012
a(n) is the number of n X n 0-1 matrices whose row sums and column sums are both {1,2,...,n}.
a(n) is the number of linear arrangements of 2n blocks of n different colors, 2 of each color, such that there are an even number of blocks between each pair of blocks of the same color.
(End)
Number of ways to place n instances of a digit inside an n X n X n cube so that no two instances lie on a plane parallel to a face of the cube (see Khovanova link, Lemma 6, p. 22). - Tanya Khovanova and Wayne Zhao, Oct 17 2018
Number of permutations P of length 2n which maximize Sum_{i=1..2n} |P_i - i|. - Fang Lixing, Dec 07 2018
REFERENCES
Archimedeans Problems Drive, Eureka, 22 (1959), 15.
David Burton, "The History of Mathematics", Sixth Edition, Problem 2, p. 433.
J. Dezert, editor, Smarandacheials, Mathematics Magazine, Aurora, Canada, No. 4/2004 (to appear).
S. M. Kerawala, The enumeration of the Latin rectangle of depth three by means of a difference equation, Bull. Calcutta Math. Soc., 33 (1941), 119-127.
J. Riordan, Combinatorial Identities, Wiley, 1968, p. 217.
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).
F. Smarandache, Back and Forth Factorials, Arizona State Univ., Special Collections, 1972.
R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 5.62(b).
LINKS
Rob Pratt (Proposer), Problem 11573, Amer. Math. Monthly, 120 (2013), 372.
Simone Severini, Title? [dead link]
FORMULA
Integral representation as n-th moment of a positive function on a positive half-axis, in Maple notation: a(n)=int(x^n*2*BesselK(0, 2*sqrt(x)), x=0..infinity), n=0, 1... - Karol A. Penson, Oct 09 2001
a(n) ~ 2*Pi*n*e^(-2*n)*n^(2*n). - Joe Keane (jgk(AT)jgk.org), Jun 07 2002
a(n) = polygorial(n, 4) = A000142(n)/ A000079(n)* A000165(n) = (n!/2^n)*Product_{i=0..n-1} (2*i + 2) = n!*Pochhammer(1, n) = n!^2. - Daniel Dockery (peritus(AT)gmail.com), Jun 13 2003
a(n) = !n!_1 = !n! = Product_{i=0, 1, 2, ... .}_{0 < |n-i| <= n}(n-i) = n(n-1)(n-2)...(2)(1)(-1)(-2)...(-n+2)(-n+1)(-n) = [(-1)^n][(n!)^2]. - J. Dezert (Jean.Dezert(AT)onera.fr), Mar 21 2004
A(x) = Sum_{n>=0,N) a(n)*x^n = 1 + x/(U(0;N-2)-x); N >= 4; U(k)= 1 + x*(k+1)^2 - x*(k+2)^2/G(k+1); besides U(0;infinity)=x; (continued fraction, Euler's 1st kind, 1-step).
Let B(x) = Sum_{n>=0} a(n)*x^n/((n!)*(n+s)!), then B(0) = 1/(1-x) for abs(x) < 1 and B(1)= -1/x * log(1-x) for abs(x)< 1.
(End).
G.f.: 1 + x*(G(0) - 1)/(x-1) where G(k) = 1 - (k+1)^2*(1 - x*G(k+1)). - Sergei N. Gladkovskii, Jan 15 2013
a(n) = det(S(i+2,j), 1 <= i,j <= n), where S(n,k) are Stirling numbers of the second kind. - Mircea Merca, Apr 04 2013
a(n) = (2*n+1)!*2^(-4*n)*Sum_{k=0..n} (-1)^k*C(2*n+1,n-k)/(2*k+1). - Mircea Merca, Nov 12 2013
a(n) = a(n-1) + 2*((n-1)^2)*sqrt(a(n-1)*a(n-2)) + ((n-1)^4)*a(n-2), for n > 1.
a(n) = a(n-1) - 2*(n^2 - 1)*sqrt(a(n-1)*a(n-2)) + (n^2 - 1)*a(n-2), for n > 1.
(End).
Sum_{n>=0} (-1)^n/a(n) = BesselJ(0,2) = A091681. (End)
Sum_{n>=0} a(n)/(2*n+1)! = 2*Pi/sqrt(27). - Daniel Suteu, Feb 06 2017
EXAMPLE
Consider the square array
1, 2, 3, 4, 5, 6, ...
2, 4, 6, 8, 10, 12, ...
3, 6, 9, 12, 15, 18, ...
4, 8, 12, 16, 20, 24, ...
5, 10, 15, 20, 25, 30, ...
...
a(3) = 36 since there are 36 functions f:[3]->[6] such that, if round(sqrt(2f(x))) = round(sqrt(2f(y))), then x=y. The functions, denoted by <f(1),f(2),f(3)>, are <1,2,4>, <1,2,5>, <1,2,6>, <1,3,4>, <1,3,5>, <1,3,6> and their respective permutations. - Dennis P. Walsh, Nov 26 2012
1 + x + 4*x^2 + 36*x^3 + 576*x^4 + 14400*x^5 + 518400*x^6 + ...
MATHEMATICA
Join[{1}, Table[Det[DiagonalMatrix[Range[n]^2]], {n, 20}]] (* Harvey P. Dale, Mar 31 2020 *)
PROG
(Haskell)
import Data.List (genericIndex)
a001044 n = genericIndex a001044_list n
a001044_list = 1 : zipWith (*) (tail a000290_list) a001044_list
(Python) import math
for n in range(0, 20): print(math.factorial(n)**2, end=', ') # Stefano Spezia, Oct 29 2018
CROSSREFS
Cf. A000142, A000292, A084939, A084940, A084941, A084942, A084943, A084944, A020549, A046032, A048617.
First right-hand column of triangle A008955.
Numbers k such that (k!)^2 + 1 is prime.
+10
20
0, 1, 2, 3, 4, 5, 9, 10, 11, 13, 24, 65, 76
COMMENTS
a(14) > 2500. - Gabriel Cunningham (gcasey(AT)mit.edu), Feb 23 2004
REFERENCES
H. Ibstedt, A Few Smarandache Sequences, Smarandache Notions Journal, Vol. 8, No. 1-2-3, 1997, 170-183.
LINKS
Eric Weisstein's World of Mathematics, Factorial
EXAMPLE
9 is a term because (9!)^2 + 1 is prime.
PROG
(Magma) [n: n in [0..90] |IsPrime(Factorial(n)^2+1)]; // Vincenzo Librandi, May 28 2015
Primes of the form (n!)^2 + 1.
+10
4
2, 5, 37, 577, 14401, 131681894401, 13168189440001, 1593350922240001, 38775788043632640001, 384956219213331276939737002152967117209600000001
EXAMPLE
37 is a term because it is prime and is (3!)^2 + 1.
PROG
(Magma) [a: n in [1..50] | IsPrime(a) where a is Factorial(n)^2+1]; // Vincenzo Librandi, Dec 08 2011
2, 2, 5, 217, 331777, 24883200001, 139314069504000001, 82606411253903523840000001, 6984964247141514123629140377600000001, 109110688415571316480344899355894085582848000000001, 395940866122425193243875570782668457763038822400000000000000000001
FORMULA
a(n) ~ (2*Pi)^(n/2) * n^(n^2 + n/2) / exp(n^2 - 1/12). - Vaclav Kotesovec, Mar 19 2018
PROG
(GAP) List([0..11], n->Factorial(n)^n+1); # Muniru A Asiru, Mar 19 2018
Smaller factor of the n-th semiprime of the form (m!)^2 + 1.
+10
3
13, 101, 17, 101, 1344169, 149, 9049, 37, 710341, 2122590346576634509, 171707860473207588349837, 7686544942807799800864250520468090636146175134909, 2196283505473, 598350346949, 1211221552894876996541369232623365900407018851538797
FORMULA
Numbers p such that p*q = ( A083340(n)!)^2 + 1, p, q prime, p < q.
EXAMPLE
a(1) = 13 because ( A083340(1)!)^2 + 1 = 518401 = 13*39877.
a(15) = 1211221552894876996541369232623365900407018851538797 because ( A083340(15)!)^2 + 1 = (55!)^2 + 1 can be factored into P52*P96 with a(15) = P52.
PROG
(PARI) for(n=1, 29, my(f=(n!)^2+1); if(bigomega(f)==2, print1(vecmin(factor(f)[, 1]), ", "))) \\ Hugo Pfoertner, Jul 13 2019
EXTENSIONS
The 11th term of the sequence (49-digit factor of the 100-digit number (41!)^2+1) was found with Yuji Kida's multiple polynomial quadratic sieve UBASIC PPMPQS v3.5 in 13 days CPU time on an Intel PIII 550 MHz.
Missing a(4) and new a(14), a(15) added by Hugo Pfoertner, Jul 13 2019
a(n) = n! * Sum_{d|n} (n/d)! / d!.
+10
2
1, 5, 37, 601, 14401, 520801, 25401601, 1626189601, 131682257281, 13168407228481, 1593350922240001, 229442707280223361, 38775788043632640001, 7600054676241325858561, 1710012252750418295078401, 437763137119219420513804801, 126513546505547170185216000001
FORMULA
E.g.f.: Sum_{k>0} k! * (exp(x^k) - 1).
If p is prime, a(p) = 1 + (p!)^2 = A020549(p).
MATHEMATICA
a[n_] := n! * DivisorSum[n, (n/#)! / #! &]; Array[a, 17] (* Amiram Eldar, Aug 30 2023 *)
PROG
(PARI) a(n) = n!*sumdiv(n, d, (n/d)!/d!);
(PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(sum(k=1, N, k!*(exp(x^k)-1))))
(Python)
from math import factorial
from sympy import divisors
f = factorial(n)
return sum(f*(a := factorial(n//d))//(b:= factorial(d)) + (f*b//a if d**2 < n else 0) for d in divisors(n, generator=True) if d**2 <= n) # Chai Wah Wu, Jun 09 2022
Number of prime factors of 1+(n!)^2 (with multiplicity).
+10
0
1, 1, 1, 1, 1, 1, 2, 2, 2, 1, 1, 1, 2, 1, 3, 2, 2, 2, 2, 2, 5, 3, 4, 3, 1, 3, 5, 7, 2, 2, 6, 4, 4, 5, 4, 3, 3, 3, 4, 4, 4, 2, 5, 5, 7, 2, 3, 8, 9, 3, 3, 4, 3, 2, 4, 2, 8, 6, 4, 3, 8, 2, 5, 4, 5, 1, 5, 5, 4, 5, 4
MATHEMATICA
Table[Length[FactorInteger[1 + (n!)^2]], {n, 0, 20}] (* T. D. Noe, Oct 17 2011 *)
AUTHOR
Charles T. Le (charlestle(AT)yahoo.com)
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