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Search: id:A014963
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| A014963 |
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a(n) = 1 unless n is a prime or prime power when a(n) = the prime in question (exponential of Mangoldt function M(n), which is log(p) if n=p^k otherwise 0). |
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42
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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
(list; graph; listen)
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OFFSET
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1,2
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COMMENT
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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 (benoit7848c(AT)orange.fr), 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 (benoit7848c(AT)orange.fr), 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 (franz.vrabec(AT)planetuniqa.at), 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 (mgranvik(AT)abo.fi), Jan 22 2008
A140580(n) = n*a(n) = (n^2)/A048671(n) = A140579 * [1,2,3,...]. - Gary W. Adamson (qntmpkt(AT)yahoo.com), May 17 2008
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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]. [From Peter Luschny (peter(AT)luschny.de), Aug 08 2009]
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LINKS
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T. D. Noe, Table of n, a(n) for n=1..10000
Leroy Quet, Home Page (listed in lieu of email address)
Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.
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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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 (vladeta(AT)eunet.rs), 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 (pbarry(AT)wit.ie), Apr 15 2005
sum_(n=1,2,3..infinity) (log(a(n))-1)/n = -2*A001620 [Bateman Manuscript Project Vol III, ed. by Erdelyi et al.] - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Mar 09 2008
a(n) = (2*Pi)^phi(n) / prod{gcd(n,k)=1} Gamma(k/n)^2 (for n > 1). [From Peter Luschny (peter(AT)luschny.de), Aug 08 2009]
a(n)=A166140/A166142. [From Mats Granvik (mats.granvik(AT)abo.fi), Oct 08 2009]
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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 (peter(AT)luschny.de), Jun 23 2009]
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CROSSREFS
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Cf. A003418. Apart from initial 1, same as A020500.
Cf. A008683, A008472, A007947, A081386, A081387.
A100994(n)=a(n)^A100995(n).
Equals row sums of triangle A140581
Cf. A140580, A048671, A140579.
First column of A140256. [From Mats Granvik (mats.granvik(AT)abo.fi), Oct 08 2009]
Sequence in context: A014973 A157753 A020500 this_sequence A099636 A099635 A086847
Adjacent sequences: A014960 A014961 A014962 this_sequence 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 (mlb(AT)well.com)
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EXTENSIONS
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Additional reference from Eric Weisstein (eric(AT)weisstein.com)
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