|
%I #38 Apr 28 2023 08:20:45
%S 1,3,5,10,9,15,13,28,33,27,21,50,25,39,45,88,33,99,37,90,65,63,45,140,
%T 145,75,153,130,57,135,61,240,105,99,117,330,73,111,125,252,81,195,85,
%U 210,297,135,93,440,385,435,165,250,105,459,189,364,185,171,117,450,121
%N Sum of gcd(k^2,n) for 1 <= k <= n.
%C a(n) is the number of non-congruent solutions to x^2*y = 0 mod n. - Yuval Dekel (dekelyuval(AT)hotmail.com), Oct 17 2003
%C Row sums of triangle A245717. - _Reinhard Zumkeller_, Jul 30 2014
%H Amiram Eldar, <a href="/A078430/b078430.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..1000 from Reinhard Zumkeller)
%H E. Krätzel, W. G. Nowak, and L. Tóth, <a href="https://eudml.org/doc/269485">On certain arithmetic functions involving the greatest common divisor</a>, Cent. Eur. J. Math., 10 (2012), 761-774.
%H M. Kühleitner and W. G. Nowak, <a href="http://arxiv.org/abs/1204.1146">On a question of A. Schinzel: Omega estimates for a special type of arithmetic functions </a>, arXiv: 1204.1146 [math.NT], 2012.
%H László Tóth, <a href="http://www.seminariomatematico.polito.it/rendiconti/69-1/97.pdf">Menon's identity and arithmetical sums representing functions of several variables</a>, Rend. Sem. Mat. Univ. Politec. Torino, 69 (2011), 97-110.
%F a(n) is multiplicative. G.f. for a(p^n), p a prime, is given by (1+(p-1)*x-p^2*x^2)/(1-p*x)/(1-p^3*x^2).
%F a(n) = n*Sum_{d|n} phi(d)*N(d)/d, where phi is Euler's totient function A000010 and N(n) is sequence A000188. - _Laszlo Toth_, Apr 15 2012
%F Multiplicative with a(p^e) = p^(3*e/2) + p^(3*e/2-1) - p^(e-1) if e is even, and 2*p^((3*e-1)/2) - p^(e-1) if e is odd. - _Amiram Eldar_, Apr 28 2023
%t Table[Sum[GCD[k^2,n],{k,n}],{n,70}] (* _Harvey P. Dale_, Sep 29 2014 *)
%t f[p_, e_] := If[EvenQ[e], p^(3*e/2) + p^(3*e/2 - 1), 2*p^((3*e - 1)/2)] - p^(e - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* _Amiram Eldar_, Apr 28 2023 *)
%o (Haskell)
%o a078430 = sum . a245717_row -- _Reinhard Zumkeller_, Jul 30 2014
%o (PARI) a(n) = sum(k=1,n, gcd(k^2, n)); \\ _Michel Marcus_, Aug 03 2016
%Y Cf. A018804, A069097, A069193.
%Y Cf. A245717.
%K mult,nonn
%O 1,2
%A _Vladeta Jovovic_, Dec 30 2002
|
