A062355 a(n) = d(n) * phi(n), where d(n) is the number of divisors function.
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
Offset: 1
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).
Crossrefs
Programs
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Magma
[NumberOfDivisors(n)*EulerPhi(n):n in [1..65]]; // Marius A. Burtea, Nov 14 2019
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Maple
seq(tau(n)*phi(n), n=1..64); # Zerinvary Lajos, Jan 22 2007
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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 *) -
PARI
a(n)=numdiv(n)*eulerphi(n); vector(150,n,a(n))
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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
Formula
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)
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