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A053819
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a(n) = Sum_{k=1..n, gcd(n,k) = 1} k^3.
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10
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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
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OFFSET
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1,3
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COMMENTS
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Except for a(2) = 1, a(n) is always divisible by n. - Jianing Song, Jul 13 2018
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REFERENCES
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Tom M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 48, problem 15, the function phi_3(n).
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LINKS
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FORMULA
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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
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MAPLE
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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:
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MATHEMATICA
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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 *)
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PROG
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(PARI) a(n) = sum(k=1, n, k^3*(gcd(n, k)==1)); \\ Michel Marcus, Mar 03 2015
(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
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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