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A014963
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Exponential of Mangoldt function M(n): a(n) = 1 unless n is a prime or prime power when a(n) = that prime.
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68
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1, 2, 3, 2, 5, 1, 7, 2, 3, 1, 11, 1, 13, 1, 1, 2, 17, 1, 19, 1, 1, 1, 23, 1, 5, 1, 3, 1, 29, 1, 31, 2, 1, 1, 1, 1, 37, 1, 1, 1, 41, 1, 43, 1, 1, 1, 47, 1, 7, 1, 1, 1, 53, 1, 1, 1, 1, 1, 59, 1, 61, 1, 1, 2, 1, 1, 67, 1, 1, 1, 71, 1, 73, 1, 1, 1, 1, 1, 79, 1, 3, 1, 83, 1, 1, 1, 1, 1, 89, 1, 1, 1, 1, 1, 1
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OFFSET
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1,2
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COMMENTS
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a(n) = gcd( C(n+1,1), C(n+2,2), ..., C(2n,n) ) where C(n,k) = binomial(n,k). - Benoit Cloitre, Jan 31 2003
a(n) = gcd(C(n,1), C(n+1,2), C(n+2,3), ...., C(2n-2,n-1)), where C(n,k) = binomial(n,k). - Benoit Cloitre, Jan 31 2003; corrected by Ant King, Dec 27 2005
Note: a(n) != GCD(A008472(n), A007947(n)), GCD of rad(n) and sopf(n) (this fails for the first time at n=30), since a(30) = 1 but GCD(rad(30), sopf(30)) = GCD(30,10) = 10.
There are arbitrarily long runs of ones (Sierpiński). - Franz Vrabec, Sep 26 2005
a(n) is the smallest positive integer such that n divides product{k=1 to n} a(k), for all positive integers n. - Leroy Quet, May 01 2007
A140580(n) = n*a(n) = n^2/A048671(n) = A140579 * [1,2,3,...]. - Gary W. Adamson, May 17 2008
Dirichlet g.f: Sum(n>0, e^Lambda(n)/n^s) = Zeta(s) + Sum(p prime, Sum(k>0, (p-1)/p^(k*s)) = Zeta(s) - ppzeta(s) + Sum(p prime, p/(p^s-1)), for a ppzeta definition see A010055. - Enrique Pérez Herrero, Jan 19 2013
Resultant of the n-th cyclotomic polynomial with the 1st cyclotomic polynomial x-1. - Ralf Stephan, Aug 14 2013
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REFERENCES
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G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 5th ed., Oxford Univ. Press, 1979, Section 17.7.
Sierpiński, W., On the numbers [1,2,...n], (Polish) Wiadom. Mat. (2) 9 1966 9-10.
Vardi, I. Computational Recreations in Mathematica. Reading, MA: Addison-Wesley, pp. 146-147, 152-153 and 249, 1991.
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LINKS
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T. D. Noe, Table of n, a(n) for n=1..10000
Peter Luschny and Stefan Wehmeier, The lcm(1,2,...,n) as a product of sine values sampled over the points in Farey sequences, arXiv:0909.1838 [math.CA], 2009.
Greg Martin, A product of Gamma function values at fractions with the same denominator, arXiv:0907.4384 [math.CA], 2009.
Carl McTague, On the Greatest Common Divisor of C(q*n,n), C(q*n,2*n), ...C(q*n,q*n-q), arXiv:1510.06696 [math.CO], 2015.
Eric Weisstein's World of Mathematics, Mangoldt Function
Eric Weisstein's World of Mathematics, Sylvester Cyclotomic Number
Index entries for sequences related to lcm's
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FORMULA
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a(n) = lcm {1..n} / lcm {1..n-1}.
a(n) = 1/Product_{d|n} d^mu(d) = Product_{d|n} (n/d)^mu(d). - Vladeta Jovovic, Jan 24 2002
a(n) = Product_{k=1..n-1, if(gcd(n, k)=1, 1-exp(2*Pi*I*k/n), 1)}, I=sqrt(-1); a(n) = n/A048671(n). - Paul Barry, Apr 15 2005
Sum_{n>=1} (log(a(n))-1)/n = -2*A001620 [Bateman Manuscript Project Vol III, ed. by Erdelyi et al.]. - R. J. Mathar, Mar 09 2008
a(n) = (2*Pi)^phi(n) / Product_{gcd(n,k)=1} Gamma(k/n)^2 (for n > 1). - Peter Luschny, Aug 08 2009
a(n) = A166140(n) / A166142(n). - Mats Granvik, Oct 08 2009
a(n) = GCD of rows in A167990. - Mats Granvik, Nov 16 2009
a(n) = A010055(n)*(A007947(n) - 1) + 1. - Reinhard Zumkeller, Mar 26 2010
a(n) = 1 + (A007947(n)-1) * floor(1/A001221(n)), for n>1. - Enrique Pérez Herrero, Jun 01 2011
a(n) = Product_{0<k<n} if(gcd(k,n)=1, 2*sin(Pi*k/n), 1). - Peter Luschny, Jun 09 2011
a(n) = exp(Sum_{k>=1} A191898(n,k)/k) for n>1 (conjecture). - Mats Granvik, Jun 19 2011
a(n) = exp(limit of zeta(s)*(Sum_{d|n} moebius(d)/d^(s-1)) as s->1) for n>1. - Mats Granvik, Jul 31 2013
a(n) = gcd{k=1..n-1} binomial(n,k), see A014410. - Michel Marcus, Dec 08 2015
a(n) = 1 + Sum_{k=2..n} (k-1)*A010051(k)*(floor(k^n/n)-floor((k^n -1)/n)). - Anthony Browne, Jun 16 2016
The Dirichlet series for log(a(n)) = Lambda(n) is given by the logarithmic derivative of the zeta function -zeta'(s)/zeta(s). - Mats Granvik, Oct 30 2016
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MAPLE
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a := n -> if n < 2 then 1 else numtheory[factorset](n); if 1 < nops(%) then 1 else op(%) fi fi; # Peter Luschny, Jun 23 2009
A014963 := n -> n/ilcm(op(numtheory[divisors](n) minus {1, n}));
seq(A014963(i), i=1..69); # Peter Luschny, Mar 23 2011
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MATHEMATICA
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a[n_?PrimeQ] := n; a[n_/; Length[FactorInteger[n]] == 1] := FactorInteger[n][[1]][[1]]; a[n_] := 1; Table[a[n], {n, 95}] (* Alonso del Arte, Jan 16 2011 *)
a[n_] := Exp[ MangoldtLambda[n]]; Table[a[n], {n, 95}] (* Jean-François Alcover, Jul 29 2013 *)
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PROG
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(PARI)
A014963(n)=
{
local(r);
if( isprime(n), return(n));
if( ispower(n, , &r) && isprime(r), return(r) );
return(1);
} \\ Joerg Arndt, Jan 16 2011
(PARI) a(n)=ispower(n, , &n); if(isprime(n), n, 1) \\ Charles R Greathouse IV, Jun 10 2011
(PARI) a(n) = gcd(vector(n-1, k, binomial(n, k))); \\ Michel Marcus, Dec 08 2015
(Haskell)
a014963 1 = 1
a014963 n | until ((> 0) . (`mod` spf)) (`div` spf) n == 1 = spf
| otherwise = 1
where spf = a020639 n
-- Reinhard Zumkeller, Sep 09 2011
(Sage)
def A014963(n) : return simplify(exp(add(moebius(d)*log(n/d) for d in divisors(n))))
[A014963(n) for n in (1..50)] # Peter Luschny, Feb 02 2012
(Python)
from sympy import factorint
def A014963(n):
....y = factorint(n)
....return list(y.keys())[0] if len(y) == 1 else 1
# Chai Wah Wu, Sep 04 2014
(cloud.sagemath)
# Choose a value for n, example with 13.
n=13
prod([1 - exp( 2*pi*I*k/n ) for k in range(n+1) if gcd(n, k) == 1]).numerical_approx();
# ".numerical_approx()" can be deleted to see the whole expression
# Eric Desbiaux, Oct 26 2014
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CROSSREFS
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Cf. A003418. Apart from initial 1, same as A020500.
Cf. A008683, A008472, A007947, A081386, A081387, A140255 (inverse Mobius transform), A140254 (Mobius transform).
A100994(n) = a(n)^A100995(n).
Equals row sums of triangle A140581.
Cf. A140580, A048671, A140579.
First column of A140256.
Cf. A120007.
Partial sum is A072107.
Sequence in context: A276835 A020500 * A157753 A099636 A099635 A178380
Adjacent sequences: A014960 A014961 A014962 * A014964 A014965 A014966
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KEYWORD
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nonn,easy,nice
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AUTHOR
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Marc LeBrun
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EXTENSIONS
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Additional reference from Eric W. Weisstein, Jun 29 2008
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STATUS
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approved
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