A053819 - OEIS

%I #46 Jun 21 2024 13:19:56

%S 1,1,9,28,100,126,441,496,1053,1100,3025,1800,6084,4410,7200,8128,

%T 18496,8910,29241,16400,29106,27830,64009,27936,77500,54756,88209,

%U 67032,164836,52200,216225,130816,185130,161840,264600,140616,443556

%N a(n) = Sum_{k=1..n, gcd(n,k) = 1} k^3.

%C Except for a(2) = 1, a(n) is always divisible by n. - _Jianing Song_, Jul 13 2018

%D Tom M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 48, problem 15, the function phi_3(n).

%H Robert Israel, <a href="/A053819/b053819.txt">Table of n, a(n) for n = 1..10000</a>

%H John D. Baum, <a href="https://www.jstor.org/stable/2690056">A Number-Theoretic Sum</a>, Mathematics Magazine 55.2 (1982): 111-113.

%H P. G. Brown, <a href="http://www.jstor.org/stable/3621931">Some comments on inverse arithmetic functions</a>, Math. Gaz. 89 (2005) 403-408.

%H Geoffrey B. Campbell, <a href="https://ideas.repec.org/a/hin/jijmms/728942.html">Dirichlet summations and products over primes</a>, Int. J. Math. Math. Sci. 16 92) (1993) 359. eq. (3.1)

%H Constantin M. Petridi, <a href="https://arxiv.org/abs/1612.07632">The Sums of the k-powers of the Euler set and their connection with Artin's conjecture for primitive roots</a>, arXiv:1612.07632 [math.NT], 2016-2018.

%F a(n) = n^2/4*(n*A000010(n) + A023900(n)), n > 1. - _Vladeta Jovovic_, Apr 17 2002

%F a(n) = eulerphi(n)*(n^3 + (-1)^omega(n)*rad(n)*n)/4. See Petridi link. - _Michel Marcus_, Jan 29 2017

%F G.f. A(x) satisfies: A(x) = x*(1 + 4*x + x^2)/(1 - x)^5 - Sum_{k>=2} k^3 * A(x^k). - _Ilya Gutkovskiy_, Mar 29 2020

%F Sum_{k=1..n} a(k) ~ 3 * n^5 / (10*Pi^2). - _Amiram Eldar_, Dec 03 2023

%p f:= proc(n) local F,t;

%p   F:= ifactors(n)[2];

%p   numtheory:-phi(n)*(n^3 + (-1)^nops(F)*mul(t[1],t=F)*n)/4

%p end proc:

%p f(1):= 1:

%p map(f, [$1..100]); # _Robert Israel_, Jan 29 2018

%t Table[Sum[j^3, {j, Select[Range[n], GCD[n, #] == 1 &]}], {n, 1, 37}] (* _Geoffrey Critzer_, Mar 03 2015 *)

%t a[1] = 1; a[n_] := Module[{f = FactorInteger[n], p, e}, p = f[[;; , 1]]; e = f[[;; , 2]]; (n^2/4) * (n * Times @@ ((p - 1)*p^(e - 1)) + Times @@ (1 - p))]; Array[a, 100] (* _Amiram Eldar_, Dec 03 2023 *)

%o (PARI) a(n) = sum(k=1,n, k^3*(gcd(n,k)==1)); \\ _Michel Marcus_, Mar 03 2015

%o (PARI) a(n) = {my(f = factor(n)); if(n == 1, 1, (n^2/4) * (n * eulerphi(f) + prod(i = 1, #f~, 1 - f[i, 1])));} \\ _Amiram Eldar_, Dec 03 2023

%Y Cf. A000010, A001221, A007947, A023900, A053818, A053820.

%Y Column k=3 of A308477.

%K nonn

%O 1,3

%A _N. J. A. Sloane_, Apr 07 2000