A053819 a(n) = Sum_{k=1..n, gcd(n,k) = 1} k^3.
1, 1, 9, 28, 100, 126, 441, 496, 1053, 1100, 3025, 1800, 6084, 4410, 7200, 8128, 18496, 8910, 29241, 16400, 29106, 27830, 64009, 27936, 77500, 54756, 88209, 67032, 164836, 52200, 216225, 130816, 185130, 161840, 264600, 140616, 443556
Offset: 1
Keywords
References
- Tom M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 48, problem 15, the function phi_3(n).
Links
- Robert Israel, Table of n, a(n) for n = 1..10000
- 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 (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.
Programs
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Maple
f:= proc(n) local F,t; F:= ifactors(n)[2]; numtheory:-phi(n)*(n^3 + (-1)^nops(F)*mul(t[1],t=F)*n)/4 end proc: f(1):= 1: map(f, [$1..100]); # Robert Israel, Jan 29 2018
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Mathematica
Table[Sum[j^3, {j, Select[Range[n], GCD[n, #] == 1 &]}], {n, 1, 37}] (* Geoffrey Critzer, Mar 03 2015 *) 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 *) -
PARI
a(n) = sum(k=1,n, k^3*(gcd(n,k)==1)); \\ Michel Marcus, Mar 03 2015
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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
Formula
a(n) = eulerphi(n)*(n^3 + (-1)^omega(n)*rad(n)*n)/4. See Petridi link. - Michel Marcus, Jan 29 2017
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
Sum_{k=1..n} a(k) ~ 3 * n^5 / (10*Pi^2). - Amiram Eldar, Dec 03 2023
Comments