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A014963 Exponential of Mangoldt function M(n): a(n) = 1 unless n is a prime or prime power when a(n) = that prime. 54
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

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

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 1-st cyclotomic polynomial x-1. - Ralf Stephan, Aug 14 2013

REFERENCES

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.

LINKS

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

Greg Martin, A product of Gamma function values at fractions with the same denominator, arXiv:0907.4384v1 [math.CA].

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

FORMULA

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). [Peter Luschny, Aug 08 2009]

a(n)=A166140(n)/A166142(n). [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. [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 divides n} moebius(d)/d^(s-1)) as s->1) for n>1. [Mats Granvik, Jul 31 2013]

MAPLE

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

MATHEMATICA

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, 1, 95}] (* Jean-François Alcover, Jul 29 2013 *)

PROG

(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

(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

CROSSREFS

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.

Cf. A120007.

Partial sum is A072107.

Sequence in context: A014973 A020500 * A157753 A099636 A099635 A178380

Adjacent sequences:  A014960 A014961 A014962 * A014964 A014965 A014966

KEYWORD

nonn,easy,nice,changed

AUTHOR

Marc LeBrun

EXTENSIONS

Additional reference from Eric W. Weisstein

STATUS

approved

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Last modified April 25 00:50 EDT 2014. Contains 240991 sequences.