OFFSET
1,3
COMMENTS
Equals row sums of triangle A143612. - Gary W. Adamson, Aug 27 2008
a(n) = A175505(n) * A023896(n) / A175506(n). For number n >= 1 holds B(n) = a(n) / A023896(n) = A175505(n) / A175506(n), where B(n) = antiharmonic mean of numbers k such that GCD(k, n) = 1 for k < n. - Jaroslav Krizek, Aug 01 2010
n does not divide a(n) iff n is a term in A316860, that is, either n is a power of 2 or n is a multiple of 3 and no prime factor of n is congruent to 1 mod 3. - Jianing Song, Jul 16 2018
REFERENCES
Tom M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 48, problem 15, the function phi_2(n).
Louis Comtet, Advanced Combinatorics, Reidel, 1974, p. 199, #2.
LINKS
G. C. Greubel, Table of n, a(n) for n = 1..2500
John D. Baum, A Number-Theoretic Sum, Mathematics Magazine 55.2 (1982): 111-113.
P. G. Brown, Some comments on inverse arithmetic functions, Math. Gaz. 89 (516) (2005) 403-408.
Geoffrey B. Campbell, Dirichlet summations and products over primes, Int. J. Math. Math. Sci. 16 92) (1993) 359. eq. (3.1)
Constantin M. Petridi, The Sums of the k-powers of the Euler set and their connection with Artin's conjecture for primitive roots, arXiv:1612.07632 [math.NT], 2016-2018.
FORMULA
If n = p_1^e_1 * ... *p_r^e_r then a(n) = n^2*phi(n)/3 + (-1)^r*p_1*..._p_r*phi(n)/6.
a(n) = A000010(n)/3 * (n^2 + (-1)^A001221(n)*A007947(n)/2)) for n>=2. - Jaroslav Krizek, Aug 24 2010
G.f. A(x) satisfies: A(x) = x*(1 + x)/(1 - x)^4 - Sum_{k>=2} k^2 * A(x^k). - Ilya Gutkovskiy, Mar 29 2020
Sum_{k=1..n} a(k) ~ n^4 / (2*Pi^2). - Amiram Eldar, Dec 03 2023
MAPLE
A053818 := proc(n)
local a, k;
a := 0 ;
for k from 1 to n do
if igcd(k, n) = 1 then
a := a+k^2 ;
end if;
end do:
a ;
end proc: # R. J. Mathar, Sep 26 2013
MATHEMATICA
a[n_] := Plus @@ (Select[ Range@n, GCD[ #, n] == 1 &]^2); Array[a, 43] (* Robert G. Wilson v, Jul 01 2010 *)
a[1] = 1; a[n_] := Module[{f = FactorInteger[n], p, e}, p = f[[;; , 1]]; e = f[[;; , 2]]; (n^2/3) * Times @@ ((p - 1)*p^(e - 1)) + (n/6) * Times @@ (1 - p)]; Array[a, 100] (* Amiram Eldar, Dec 03 2023 *)
PROG
(PARI) a(n) = sum(k=1, n, k^2*(gcd(n, k) == 1)); \\ Michel Marcus, Jan 30 2016
(PARI) a(n) = {my(f = factor(n)); if(n == 1, 1, (n^2/3) * eulerphi(f) + (n/6) * prod(i = 1, #f~, 1 - f[i, 1])); } \\ Amiram Eldar, Dec 03 2023
CROSSREFS
KEYWORD
nonn
AUTHOR
N. J. A. Sloane, Apr 07 2000
STATUS
approved