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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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51
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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 (Sierpinski). - 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
a(n)*A100994 gives the last row of the columns in A133233. - Mats Granvik, Jan 22 2008
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
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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.
Sierpi\'nski, 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.
Greg Martin, A product of Gamma function values at fractions with the same denominator, arXiv:0907.4384v1[math.CA].
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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
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 divides n } d^mu(d)=Product_{ d divides 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) / prod{gcd(n,k)=1} Gamma(k/n)^2 (for n > 1). [From Peter Luschny, Aug 08 2009]
a(n)=A166140(n)/A166142(n). [From Mats Granvik, Oct 08 2009]
a(n) = GCD of rows in A167990. [From Mats Granvik, Nov 16 2009]
a(n) = A010055(n)*(A007947(n) - 1) + 1. [From Reinhard Zumkeller, Mar 26 2010]
a(n) = 1+(A007947(n)-1)*floor(1/A001221(n))), for n>1. [From 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). [From Mats Granvik, Jun 19 2011]
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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; [From 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}] (* From Alonso del Arte, Jan 16 2011 *)
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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);
}
vector(99, n, A014963(n)) /* show terms */
/* From Joerg Arndt, Jan 16 2011 */
(PARI) a(n)=ispower(n, , &n); if(isprime(n), n, 1) \\ Charles R Greathouse IV, Jun 10 2011
(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
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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 (inv. Mobius trans.), A140254 (Mobius trans.).
A100994(n)=a(n)^A100995(n).
Equals row sums of triangle A140581
Cf. A140580, A048671, A140579.
First column of A140256. [From Mats Granvik, Oct 08 2009]
Cf. A120007. [From Reinhard Zumkeller, Mar 26 2010]
Partial sum is A072107. [From Enrique Pérez Herrero, Jun 12 2011]
Sequence in context: A014973 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,changed
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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
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STATUS
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approved
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