A014963 Exponential of Mangoldt function M(n): a(n) = 1 unless n is a prime or prime power, in which case a(n) = that prime.
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
Offset: 1
References
- G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 5th ed., Oxford Univ. Press, 1979, Section 17.7.
- I. Vardi, 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, 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.
- A. Nowicki, Strong divisibility and LCM-sequences, arXiv:1310.2416 [math.NT], 2013.
- A. Nowicki, Strong divisibility and LCM-sequences, Am. Math. Mnthly 122 (2015), 958-966.
- W. Sierpiński, On the numbers [1,2,...n], (Polish) Wiadom. Mat. (2) 9 1966 9-10.
- 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
Crossrefs
Programs
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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 -
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 # The following is Nowicki's LCM-Transform - N. J. A. Sloane, Jan 09 2024 LCMXFM:=proc(a) local p,q,b,i,k,n: if whattype(a) <> list then RETURN([]); fi: n:=nops(a): b:=[a[1]]: p:=[a[1]]; for i from 2 to n do q:=[op(p),a[i]]; k := lcm(op(q))/lcm(op(p)); b:=[op(b),k]; p:=q;; od: RETURN(b); end: # Alternative, to be called by 'seq' as shown, not for a single n. a := proc(n) option remember; local i; global f; f := ifelse(n=1, 1, f*n); iquo(f, mul(a(i)^iquo(n, i), i=1..n-1)) end: seq(a(n), n=1..95); # Peter Luschny, Apr 05 2025
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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, 95}] (* Jean-François Alcover, Jul 29 2013 *) Ratios[LCM @@ # & /@ Table[Range[n], {n, 100}]] (* Horst H. Manninger, Mar 08 2024 *) Table[Which[PrimeQ[n],n,PrimePowerQ[n],Surd[n,FactorInteger[n][[-1,2]]],True,1],{n,100}] (* Harvey P. Dale, Mar 01 2025 *) -
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
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PARI
a(n)=ispower(n,,&n);if(isprime(n),n,1) \\ Charles R Greathouse IV, Jun 10 2011
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Python
from sympy import factorint def A014963(n): y = factorint(n) return list(y.keys())[0] if len(y) == 1 else 1 print([A014963(n) for n in range(1, 71)]) # Chai Wah Wu, Sep 04 2014
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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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Sage
def a(n): if n == 1: return 1 return prod(1 - E(n)**k for k in ZZ(n).coprime_integers(n+1)) [a(n) for n in range(1, 14)] # F. Chapoton, Mar 17 2020
Formula
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.]
a(n) = 1/Product_{d|n} d^mu(d) = Product_{d|n} (n/d)^mu(d). - Vladeta Jovovic, Jan 24 2002
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)) = 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.
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) = GCD of rows in A167990. - Mats Granvik, Nov 16 2009
a(n) = Product_{k=1..n-1} 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
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
a(n) = exp(lim_{s->1} zeta(s)*Sum_{d|n} moebius(d)/d^(s-1)) for n>1. - Mats Granvik, Jul 31 2013
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]
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
a(n) = A008578(1+A297109(n)), For all n >= 1, Product_{d|n} a(d) = n. - Antti Karttunen, Feb 01 2021
Product_{k=1..floor(n/2)} Product_{j=1..floor(n/k)} a(j) = n!. - Ammar Khatab, Jan 28 2025
Extensions
Additional reference from Eric W. Weisstein, Jun 29 2008
Comments