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 A058026 Number of positive integers, k, where k <= n and gcd(k,n) = gcd(k+1,n) = 1. 20
 1, 0, 1, 0, 3, 0, 5, 0, 3, 0, 9, 0, 11, 0, 3, 0, 15, 0, 17, 0, 5, 0, 21, 0, 15, 0, 9, 0, 27, 0, 29, 0, 9, 0, 15, 0, 35, 0, 11, 0, 39, 0, 41, 0, 9, 0, 45, 0, 35, 0, 15, 0, 51, 0, 27, 0, 17, 0, 57, 0, 59, 0, 15, 0, 33, 0, 65, 0, 21, 0, 69, 0, 71, 0, 15, 0, 45, 0, 77, 0, 27, 0, 81, 0, 45, 0, 27, 0 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,5 COMMENTS Called the Schemmel totient function in the Handbook of Number Theory II. - R. J. Mathar, Apr 15 2011 a(n) is also the number of units u in Z/nZ such that Phi(1,u) or Phi(2,u) is a unit, where Phi is the cyclotomic polynomial. - Jordan Lenchitz, Jul 12 2017 This is the function phi(n, 1) in Alder. - Michel Marcus, Nov 14 2017 REFERENCES József Sándor and Borislav Crstici, Handbook of Number theory II, Kluwer Academic Publishers, 2004, Chapter 3, p. 276. LINKS Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..5000 from Enrique Pérez Herrero) Henry L. Alder, A Generalization of the Euler phi-Function, The American Mathematical Monthly, Vol. 65, No. 9 (Nov., 1958), pp. 690-692. O. Bordelles and B. Cloitre, An Alternating Sum Involving the Reciprocal of Certain Multiplicative Functions, J. Int. Seq. 16 (2013) #13.6.3. Colin Defant, On Arithmetic Functions Related to Iterates of the Schemmel Totient Functions, J. Int. Seq. 18 (2015) # 15.2.1. Leonard Eugene Dickson, Schemmel's Generalization of Euler's phi-Function, History of the Theory of Numbers, Vol. 1: Divisibility and Primality, Washington, Carnegie Institution of Washington, 1919, p. 147. Mizan R. Khan and Riaz R. Khan, To count clean triangles we count on imph(n), arXiv:2012.11081 [math.CO], 2020. Walter Klotz and Torsten Sander, Some Properties of Unitary Cayley Graphs, The Electronic Journal of Combinatorics, Volume 14 (2007), #R45. See Corollary 7 p. 4. Victor Schemmel, Ueber relative Primzahlen, Journal für die reine und angewandte Mathematik, Vol. 70 (1869), pp. 191-192. FORMULA Multiplicative with a(p^e) = p^(e-1)*(p-2). - Vladeta Jovovic, Dec 01 2001 a(n) = Sum_{d|n} n/d*mu(d)*tau(d). - Vladeta Jovovic, Apr 29 2002 a(n) = Sum_{d divides n} phi(n/d)*(-1)^omega(d). - Vladeta Jovovic, Sep 26 2002 A003557(n) | a(n). - R. J. Mathar, Mar 30 2011 a(n) = n*Product_{primes p|n} (1-2/p). Dirichlet g.f. zeta(s-1)*product_p (1-2*p^(-s)). - R. J. Mathar, Apr 15 2011 a(n) = phi(n) * Sum_{d|n} mu(d)/phi(d), where mu(k) is the Moebius function and phi(k) is the Euler totient function. - Daniel Suteu, Jun 23 2018 Sum_{k=1..n} a(k) ~ c * Pi^2 * n^2 / 2, where c = A065474 = Product_{primes p} (1 - 2/p^2) = 0.32263409893924467057953169254823706657095... - Vaclav Kotesovec, Dec 18 2019 a(n) = Sum_{k=1..n} (-1)^omega(gcd(n,k)). - Ilya Gutkovskiy, Feb 22 2020 EXAMPLE a(15) = 3 because 1 and 2, 7 and 8 and 13 and 14 are all relatively prime to 15. a(15) = a(3*5) = a(3)*a(5) = 3^0*(3-2)*5^0*(5-2) = 3. MAPLE A058026 := proc(n) local a; a := n ; for p in numtheory[factorset](n) do a := a*(1-2/p) ; end do: a ; end proc: # R. J. Mathar, Apr 15 2011 MATHEMATICA a[n_] := DivisorSum[n, n/# MoebiusMu[#] DivisorSigma[0, #]&]; Array[a, 90] (* Jean-François Alcover, Dec 05 2015 *) f[p_, e_] := (p-2) * p^(e-1); a = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, May 01 2020 *) PROG (PARI) a(n) = sumdiv(n, d, n/d*moebius(d)*numdiv(d)); \\ Michel Marcus, Apr 27 2014 (PARI) a(n) = n*prod(p=1, n, if (isprime(p) && !(n % p), (1-2/p), 1)); \\ Michel Marcus, Feb 02 2016 (Haskell) a058026 n = product \$ zipWith (\p e -> p ^ (e - 1) * (p - 2))                               (a027748_row n) (a124010_row n) -- Reinhard Zumkeller, May 10 2014 CROSSREFS Cf. A070554, A069828, A027748, A124010, A289460. Cf. A000010 (phi(n,0)), A002472 (phi(n,2)). Sequence in context: A340525 A003966 A123931 * A004605 A175919 A086664 Adjacent sequences:  A058023 A058024 A058025 * A058027 A058028 A058029 KEYWORD nonn,mult AUTHOR Leroy Quet, Nov 15 2000 STATUS approved

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Last modified May 16 22:04 EDT 2021. Contains 343955 sequences. (Running on oeis4.)