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 A106398 Binomial transform of denominators in a Zeta function. 0
 1, -1, -6, -19, -39, -66, -98, -129, -172, -330, -908, -2502, -5955, -12107 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,3 COMMENTS The formula 1/Zeta(s) = 1 - 1/2^s - 1/3^s - 1/5^s + 1/6^s is shown on p. 249 of Derbyshire and relies upon strategies pioneered by Euler. REFERENCES John Derbyshire, "Prime Obsession, Bernhard Riemann and the Greatest Unsolved Problem in Mathematics", Joseph Henry Press, 2003, p. 249. LINKS Table of n, a(n) for n=1..14. Les Mathematiques.net, Formule 1/zeta(s) (in French) FORMULA Given 1/Zeta(s) = 1 - 1/2^s - 1/3^s - 1/5^s + 1/6^s - 1/7^s + 1/10^s - 1/11^s ..., we apply the binomial transform to the terms [1, -2, -3, -5, 6, -7, 10, -11, -13, 14, 15, -17, -19, 21, ...], which is the set of squarefree numbers (A005117), along with the Mobius function of each term. EXAMPLE The terms 1, 2, 3, 5, 6, 7, ... = A005117, squarefree numbers. Applying the Mobius function rules to each of these, we get 1, -2, -3, -5, 6, .... The Mobius function rules are: Given the domain N, the natural numbers 1,2,3,..., Mu(1) = 1; Mu(n) of N = 0 if n has a square factor; Mu(n) = -1 if n is a prime or the product of an odd number of different primes; Mu(n) = 1 if n is the product of an even numbers of different primes. CROSSREFS Cf. A005117, A008683. Sequence in context: A010899 A090381 A354343 * A179986 A054567 A096957 Adjacent sequences: A106395 A106396 A106397 * A106399 A106400 A106401 KEYWORD sign,more AUTHOR Gary W. Adamson, May 01 2005 STATUS approved

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Last modified August 11 05:00 EDT 2024. Contains 375059 sequences. (Running on oeis4.)