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 A062355 a(n) = d(n) * phi(n), where d(n) is the number of divisors function. 25
 1, 2, 4, 6, 8, 8, 12, 16, 18, 16, 20, 24, 24, 24, 32, 40, 32, 36, 36, 48, 48, 40, 44, 64, 60, 48, 72, 72, 56, 64, 60, 96, 80, 64, 96, 108, 72, 72, 96, 128, 80, 96, 84, 120, 144, 88, 92, 160, 126, 120, 128, 144, 104, 144, 160, 192, 144, 112, 116, 192, 120, 120, 216, 224 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS a(n) = sum of gcd(k-1,n) for 1 <= k <= n and gcd(k,n)=1 (Menon's identity). For n = 2^(4*k^2 - 1), k >= 1, the terms of the sequence are square and for n = 2^((3*k + 2)^3 - 1), k >= 1, the terms of the sequence are cubes. - Marius A. Burtea, Nov 14 2019 Sum_{k>=1} 1/a(k) diverges. - Vaclav Kotesovec, Sep 20 2020 REFERENCES D. M. Burton, Elementary Number Theory, Allyn and Bacon Inc., Boston MA, 1976, Prob. 7.2 12, p. 141. P. K. Menon, On the sum gcd(a-1,n) [(a,n)=1], J. Indian Math. Soc. (N.S.), 29 (1965), 155-163. József Sándor, On Dedekind's arithmetical function, Seminarul de teoria structurilor (in Romanian), No. 51, Univ. Timișoara, 1988, pp. 1-15. See p. 11. József Sándor, Some diophantine equations for particular arithmetic functions (in Romanian), Seminarul de teoria structurilor, No. 53, Univ. Timișoara, 1989, pp. 1-10. See p. 8. LINKS Antti Karttunen, Table of n, a(n) for n = 1..65537 (terms 1..1000 from Harry J. Smith) Pentti Haukkanen and László Tóth, Menon-type identities again: Note on a paper by Li, Kim and Qiao, arXiv:1911.05411 [math.NT], 2019. Vaclav Kotesovec, Graph - the asymptotic ratio (250000000 terms) R. J. Mathar, Survey of Dirichlet series of multiplicative arithmetic functions, arXiv:1106.4038 [math.NT], 2011-2012, Section 3.15. R. Sivaramakrishnan, Problem E 1962, Elementary Problems, The American Mathematical Monthly, Vol. 74, No. 2 (1967), p. 198; Solution, ibid., Vol. 75, No. 5 (1968), p. 550. Marius Tarnauceanu, A generalization of the Menon's identity, arXiv:1109.2198 [math.GR], 2011-2012. Laszlo Toth, Menon's identity and arithmetical sums representing functions of several variables, Rend. Sem. Mat. Univ. Politec. Torino, 69 (2011), 97-110. Wikipedia, Arithmetic function (Menon's identity). FORMULA Dirichlet convolution of A047994 and A000010. - R. J. Mathar, Apr 15 2011 a(n) = A000005(n)*A000010(n). Multiplicative with a(p^e) = (e+1)*(p-1)*p^(e-1). - R. J. Mathar, Jun 23 2018 a(n) = A173557(n) * A318519(n) = A003557(n) * A304408(n). - Antti Karttunen, Sep 16 2018 & Sep 20 2019 From Vaclav Kotesovec, Jun 15 2020: (Start) Let f(s) = Product_{primes p} (1 - 2*p^(-s) + p^(1-2*s)). Dirichlet g.f.: zeta(s-1)^2 * f(s). Sum_{k=1..n} a(k) ~ n^2 * (f(2)*(log(n)/2 + gamma - 1/4) + f'(2)/2), where f(2) = A065464 = Product_{primes p} (1 - 2/p^2 + 1/p^3) = 0.42824950567709444..., f'(2) = 2 * A065464 * A335707 = f(2) * Sum_{primes p} 2*log(p) / (p^2 + p - 1) = 0.35866545223424232469545420783620795... and gamma is the Euler-Mascheroni constant A001620. (End) From Amiram Eldar, Mar 02 2021: (Start) a(n) >= n (Sivaramakrishnan, 1967). a(n) >= sigma(n), for odd n (Sándor, 1988). a(n) >= phi(n) + n - 1 (Sándor, 1989) (End) From Richard L. Ollerton, May 07 2021: (Start) a(n) = Sum_{k=1..n} uphi(gcd(n,k)), where uphi(n) = A047994(n). a(n) = Sum_{k=1..n} uphi(n/gcd(n,k))*phi(gcd(n,k))/phi(n/gcd(n,k)). (End) MAPLE seq(tau(n)*phi(n), n=1..64); # Zerinvary Lajos, Jan 22 2007 MATHEMATICA Table[EulerPhi[n] DivisorSigma[0, n], {n, 80}] (* Carl Najafi, Aug 16 2011 *) f[p_, e_] := (e+1)*(p-1)*p^(e-1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 21 2020 *) PROG (PARI) a(n)=numdiv(n)*eulerphi(n); vector(150, n, a(n)) (PARI) { for (n=1, 1000, write("b062355.txt", n, " ", numdiv(n)*eulerphi(n)) ) } \\ Harry J. Smith, Aug 05 2009 (PARI) for(n=1, 100, print1(direuler(p=2, n, (1 - 2*X + p*X^2)/(1 - p*X)^2)[n], ", ")) \\ Vaclav Kotesovec, Jun 15 2020 (Magma) [NumberOfDivisors(n)*EulerPhi(n):n in [1..65]]; // Marius A. Burtea, Nov 14 2019 CROSSREFS Cf. A003557, A173557, A061468, A062816, A079535, A062949 (inverse Mobius transform), A304408, A318519, A327169 (number of times n occurs in this sequence). Cf. A062354, A064840. Sequence in context: A063199 A219028 A333557 * A087671 A088308 A167832 Adjacent sequences: A062352 A062353 A062354 * A062356 A062357 A062358 KEYWORD easy,nonn,mult AUTHOR Jason Earls, Jul 06 2001 STATUS approved

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Last modified December 9 19:36 EST 2022. Contains 358703 sequences. (Running on oeis4.)