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A053819 a(n) = Sum_{k=1..n, gcd(n,k) = 1} k^3. 6
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

Except for a(2) = 1, a(n) is always divisible by n. - Jianing Song, Jul 13 2018

REFERENCES

T. 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.

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.

FORMULA

a(n) = n^2/4*(n*A000010(n) + A023900(n)), n > 1. - Vladeta Jovovic, Apr 17 2002

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

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

MATHEMATICA

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

PROG

(PARI) a(n) = sum(k=1, n, k^3*(gcd(n, k)==1)); \\ Michel Marcus, Mar 03 2015

CROSSREFS

Cf. A053818.

Sequence in context: A171215 A296601 A294567 * A294287 A085292 A198059

Adjacent sequences:  A053816 A053817 A053818 * A053820 A053821 A053822

KEYWORD

nonn

AUTHOR

N. J. A. Sloane, Apr 07 2000

STATUS

approved

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Last modified November 20 05:07 EST 2019. Contains 329323 sequences. (Running on oeis4.)