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%I #249 Mar 21 2024 14:24:54
%S 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,
%T 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,
%U 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
%N Exponential of Mangoldt function M(n): a(n) = 1 unless n is a prime or prime power, in which case a(n) = that prime.
%C There are arbitrarily long runs of ones (Sierpiński). - _Franz Vrabec_, Sep 26 2005
%C a(n) is the smallest positive integer such that n divides Product_{k=1..n} a(k), for all positive integers n. - _Leroy Quet_, May 01 2007
%C Resultant of the n-th cyclotomic polynomial with the 1st cyclotomic polynomial x-1. - _Ralf Stephan_, Aug 14 2013
%C A368749(n) is the smallest prime p such that the interval [a(p), a(q)] contains n 1's; q = nextprime(p), n >= 0. - _David James Sycamore_, Mar 21 2024
%D G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 5th ed., Oxford Univ. Press, 1979, Section 17.7.
%D I. Vardi, Computational Recreations in Mathematica. Reading, MA: Addison-Wesley, pp. 146-147, 152-153 and 249, 1991.
%H T. D. Noe, <a href="/A014963/b014963.txt">Table of n, a(n) for n = 1..10000</a>
%H Peter Luschny and Stefan Wehmeier, <a href="http://arxiv.org/abs/0909.1838">The lcm(1,2,...,n) as a product of sine values sampled over the points in Farey sequences</a>, arXiv:0909.1838 [math.CA], 2009.
%H Greg Martin, <a href="http://arxiv.org/abs/0907.4384">A product of Gamma function values at fractions with the same denominator</a>, arXiv:0907.4384 [math.CA], 2009.
%H Carl McTague, <a href="http://arxiv.org/abs/1510.06696">On the Greatest Common Divisor of C(q*n,n), C(q*n,2*n), ...C(q*n,q*n-q)</a>, arXiv:1510.06696 [math.CO], 2015.
%H A. Nowicki, <a href="http://arxiv.org/abs/1310.2416">Strong divisibility and LCM-sequences</a>, arXiv:1310.2416 [math.NT], 2013.
%H A. Nowicki, <a href="http://www.jstor.org/stable/10.4169/amer.math.monthly.122.10.958">Strong divisibility and LCM-sequences</a>, Am. Math. Mnthly 122 (2015), 958-966.
%H W. Sierpiński, <a href="https://doi.org/10.14708/wm.v9i1.2263">On the numbers [1,2,...n]</a>, (Polish) Wiadom. Mat. (2) 9 1966 9-10.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/MangoldtFunction.html">Mangoldt Function</a>.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/SylvesterCyclotomicNumber.html">Sylvester Cyclotomic Number</a>.
%H <a href="/index/Lc#lcm">Index entries for sequences related to lcm's</a>
%F a(n) = A003418(n) / A003418(n-1) = lcm {1..n} / lcm {1..n-1}. [This is equivalent to saying that this sequence is the LCM-transform (as defined by Nowicki, 2013) of the positive integers. - _David James Sycamore_, Jan 09 2024.]
%F a(n) = 1/Product_{d|n} d^mu(d) = Product_{d|n} (n/d)^mu(d). - _Vladeta Jovovic_, Jan 24 2002
%F 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
%F 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
%F Note: a(n) != gcd(A008472(n), A007947(n)) = A099636(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.
%F a(n)^A100995(n) = A100994(n). - _N. J. A. Sloane_, Feb 20 2005
%F 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
%F 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
%F n*a(n) = A140580(n) = n^2/A048671(n) = A140579 * [1,2,3,...]. - _Gary W. Adamson_, May 17 2008
%F a(n) = (2*Pi)^phi(n) / Product_{gcd(n,k)=1} Gamma(k/n)^2 (for n > 1). - _Peter Luschny_, Aug 08 2009
%F a(n) = A166140(n) / A166142(n). - _Mats Granvik_, Oct 08 2009
%F a(n) = GCD of rows in A167990. - _Mats Granvik_, Nov 16 2009
%F a(n) = A010055(n)*(A007947(n) - 1) + 1. - _Reinhard Zumkeller_, Mar 26 2010
%F a(n) = 1 + (A007947(n)-1) * floor(1/A001221(n)), for n>1. - _Enrique Pérez Herrero_, Jun 01 2011
%F a(n) = Product_{k=1..n-1} if(gcd(k,n)=1, 2*sin(Pi*k/n), 1). - _Peter Luschny_, Jun 09 2011
%F a(n) = exp(Sum_{k>=1} A191898(n,k)/k) for n>1 (conjecture). - _Mats Granvik_, Jun 19 2011
%F 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
%F a(n) = exp(lim_{x->1} zeta(s)*Sum_{d|n} moebius(d)/d^(s-1)) for n>1. - _Mats Granvik_, Jul 31 2013
%F a(n) = gcd_{k=1..n-1} binomial(n,k) for n > 1, see A014410. - _Michel Marcus_, Dec 08 2015 [Corrected by _Jinyuan Wang_, Mar 20 2020]
%F 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
%F 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
%F a(n) = A008578(1+A297109(n)), For all n >= 1, Product_{d|n} a(d) = n. - _Antti Karttunen_, Feb 01 2021
%p 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
%p A014963 := n -> n/ilcm(op(numtheory[divisors](n) minus {1,n}));
%p seq(A014963(i), i=1..69); # _Peter Luschny_, Mar 23 2011
%p # The following is Nowicki's LCM-Transform - _N. J. A. Sloane_, Jan 09 2024
%p LCMXFM:=proc(a) local p,q,b,i,k,n:
%p if whattype(a) <> list then RETURN([]); fi:
%p n:=nops(a):
%p b:=[a[1]]: p:=[a[1]];
%p for i from 2 to n do q:=[op(p),a[i]]; k := lcm(op(q))/lcm(op(p));
%p b:=[op(b),k]; p:=q;; od:
%p RETURN(b);
%p end:
%t 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 *)
%t a[n_] := Exp[ MangoldtLambda[n]]; Table[a[n], {n, 95}] (* _Jean-François Alcover_, Jul 29 2013 *)
%t Ratios[LCM @@ # & /@ Table[Range[n], {n, 100}]] (* _Horst H. Manninger_, Mar 08 2024 *)
%o (PARI)
%o A014963(n)=
%o {
%o local(r);
%o if( isprime(n), return(n));
%o if( ispower(n,,&r) && isprime(r), return(r) );
%o return(1);
%o } \\ _Joerg Arndt_, Jan 16 2011
%o (PARI) a(n)=ispower(n,,&n);if(isprime(n),n,1) \\ _Charles R Greathouse IV_, Jun 10 2011
%o (Haskell)
%o a014963 1 = 1
%o a014963 n | until ((> 0) . (`mod` spf)) (`div` spf) n == 1 = spf
%o | otherwise = 1
%o where spf = a020639 n
%o -- _Reinhard Zumkeller_, Sep 09 2011
%o (Sage)
%o def A014963(n) : return simplify(exp(add(moebius(d)*log(n/d) for d in divisors(n))))
%o [A014963(n) for n in (1..50)] # _Peter Luschny_, Feb 02 2012
%o (Sage)
%o def a(n):
%o if n == 1: return 1
%o return prod(1 - E(n)**k for k in ZZ(n).coprime_integers(n+1))
%o [a(n) for n in range(1, 14)] # _F. Chapoton_, Mar 17 2020
%o (Python)
%o from sympy import factorint
%o def A014963(n):
%o y = factorint(n)
%o return list(y.keys())[0] if len(y) == 1 else 1
%o print([A014963(n) for n in range(1, 71)]) # _Chai Wah Wu_, Sep 04 2014
%Y Apart from initial 1, same as A020500. With ones replaced by zeros, equal to A120007.
%Y Cf. A003418, A007947, A008683, A008472, A008578, A048671 (= n/a(n)), A072107 (partial sums), A081386, A081387, A099636, A100994, A100995, A140255 (inverse Mobius transform), A140254 (Mobius transform), A297108, A297109, A340675.
%Y First column of A140256.
%Y Row sums of triangle A140581. Cf. also A140579, A140580 (= n*a(n)).
%Y Cf. A000027, A348846.
%Y Cf. A368749.
%K nonn,easy,nice
%O 1,2
%A _Marc LeBrun_
%E Additional reference from _Eric W. Weisstein_, Jun 29 2008
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