|
| |
|
|
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
|
| |
|
|
