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 A059376 Jordan function J_3(n). 24

%I

%S 1,7,26,56,124,182,342,448,702,868,1330,1456,2196,2394,3224,3584,4912,

%T 4914,6858,6944,8892,9310,12166,11648,15500,15372,18954,19152,24388,

%U 22568,29790,28672,34580,34384,42408,39312,50652,48006,57096

%N Jordan function J_3(n).

%D L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 199, #3.

%D R. Sivaramakrishnan, "The many facets of Euler's totient. II. Generalizations and analogues", Nieuw Arch. Wisk. (4) 8 (1990), no. 2, 169-187.

%H T. D. Noe, <a href="/A059376/b059376.txt">Table of n, a(n) for n = 1..1000</a>

%F Multiplicative with a(p^e) = p^(3e) - p^(3e-3). - _Vladeta Jovovic_, Jul 26 2001

%F a(n) = Sum_{d|n} d^3*mu(n/d). - _Benoit Cloitre_, Apr 05 2002

%F Dirichlet generating function: zeta(s-3)/zeta(s). - _Franklin T. Adams-Watters_, Sep 11 2005

%F A063453(n) divides a(n). - _R. J. Mathar_, Mar 30 2011

%F a(n) = Sum_{k=1..n} gcd(k,n)^3 * cos(2*Pi*k/n). - _Enrique PĂ©rez Herrero_, Jan 18 2013

%F a(n) = n^3*Product_{distinct primes p dividing n} (1-1/p^3). - _Tom Edgar_, Jan 09 2015

%F G.f.: Sum_{n>=1} a(n)*x^n/(1 - x^n) = x*(1 + 4*x + x^2)/(1 - x)^4. - _Ilya Gutkovskiy_, Apr 25 2017

%F Sum_{d|n} a(d) = n^3. - _Werner Schulte_, Jan 12 2018

%p J := proc(n,k) local i,p,t1,t2; t1 := n^k; for p from 1 to n do if isprime(p) and n mod p = 0 then t1 := t1*(1-p^(-k)); fi; od; t1; end; # (with k = 3)

%p A059376 := proc(n)

%p end proc: # _R. J. Mathar_, Nov 03 2015

%t JordanJ[n_, k_: 1] := DivisorSum[n, #^k*MoebiusMu[n/#] &]; f[n_] := JordanJ[n, 3]; Array[f, 39]

%o (PARI) for(n=1,120,print1(sumdiv(n,d,d^3*moebius(n/d)),","))

%o (PARI) for (n = 1, 1000, write("b059376.txt", n, " ", sumdiv(n, d, d^3*moebius(n/d))); ) \\ _Harry J. Smith_, Jun 26 2009

%o (PARI) seq(n) = dirmul(vector(n,k,k^3), vector(n,k,moebius(k)));

%o seq(39) \\ _Gheorghe Coserea_, May 11 2016

%Y See A059379 and A059380 (triangle of values of J_k(n)), A000010 (J_1), A007434 (J_2), A059377 (J_4), A059378 (J_5).

%K nonn,mult

%O 1,2

%A _N. J. A. Sloane_, Jan 28 2001

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Last modified May 28 06:06 EDT 2018. Contains 304734 sequences. (Running on oeis4.)