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A062355 a(n) = d(n) * phi(n), where d(n) is the number of divisors function. 18
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

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.

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..65537 (terms 1..1000 from Harry J. Smith)

Pentti Haukkanen, 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.

M. 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

Numerically: Sum_{k=1..n} a(k) ~ n^2*(0.214125...*log(n) + 0.31947...). - Vaclav Kotesovec, Feb 05 2019

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 *)

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

(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 June 6 18:59 EDT 2020. Contains 334832 sequences. (Running on oeis4.)