A173290 - OEIS

%I #58 Sep 08 2022 08:45:50

%S 1,4,8,14,20,32,40,52,64,82,94,118,132,156,180,204,222,258,278,314,

%T 346,382,406,454,484,526,562,610,640,712,744,792,840,894,942,1014,

%U 1052,1112,1168,1240,1282,1378,1422,1494,1566,1638,1686,1782,1838,1928,2000,2084

%N Partial sums of A001615.

%C a(n) is even for n >= 2. - _Jianing Song_, Nov 24 2018

%D W. Hürlimann, Dedekind's arithmetic function and primitive four squares counting functions, Journal of Algebra, Number Theory: Advances and Applications, Volume 14, Number 2, 2015, Pages 73-88; http://scientificadvances.co.in; DOI: http://dx.doi.org/10.18642/jantaa_7100121599

%H Enrique Pérez Herrero, <a href="/A173290/b173290.txt">Table of n, a(n) for n = 1..5000</a>

%H W. Hürlimann, <a href="https://www.researchgate.net/publication/295616503_Dedekind&#39;s_arithmetic_function_and_primitive_four_squares_counting_functions">Dedekind's arithmetic function and primitive four squares counting functions</a>, Journal of Algebra, Number Theory: Advances and Applications, Volume 14, Number 2, 2015, Pages 73-88.

%F a(n) = Sum_{i=1..n} A001615(i) = Sum_{i=1..n} (n * Product_{p|n, p prime} (1 + 1/p)).

%F a(n) = 15*n^2/(2*Pi^2) + O(n*log(n)). - _Enrique Pérez Herrero_, Jan 14 2012

%F a(n) = Sum_{i=1..n} A063659(i) * floor(n/i). - _Enrique Pérez Herrero_, Feb 23 2013

%F a(n) = (1/2)*Sum_{k=1..n} mu(k)^2 * floor(n/k) * floor(1+n/k), where mu(k) is the Moebius function. - _Daniel Suteu_, Nov 19 2018

%F a(n) = (Sum_{k=1..floor(sqrt(n))} k*(k+1) * (A013928(1+floor(n/k)) - A013928(1+floor(n/(k+1)))) + Sum_{k=1..floor(n/(1+floor(sqrt(n))))} mu(k)^2 * floor(n/k) * floor(1+n/k))/2. - _Daniel Suteu_, Nov 23 2018

%p with(numtheory): a:=n->(1/2)*add(mobius(k)^2*floor(n/k)*floor(1+n/k),k=1..n); seq(a(n),n=1..55); # _Muniru A Asiru_, Nov 24 2018

%t Table[Sum[DirichletConvolve[j, MoebiusMu[j]^2, j, k], {k,1,n}], {n,60}] (* _G. C. Greubel_, Nov 23 2018 *)

%t psi[n_] := If[n==1, 1, n*Times@@(1 + 1/FactorInteger[n][[;;,1]])]; Accumulate[Array[psi, 50]] (* _Amiram Eldar_, Nov 23 2018 *)

%o (PARI)

%o S(n) = sum(k=1, sqrtint(n), moebius(k)*(n\(k*k))); \\ see: A013928

%o a(n) = sum(k=1, sqrtint(n), k*(k+1) * (S(n\k) - S(n\(k+1))))/2 + sum(k=1, n\(1+sqrtint(n)), moebius(k)^2*(n\k)*(1+n\k))/2; \\ _Daniel Suteu_, Nov 23 2018

%o (Sage)

%o def A173290(n) :

%o     return add(k*mul(1+1/p for p in prime_divisors(k)) for k in (1..n))

%o [A173290(n) for n in (1..52)]  # _Peter Luschny_, Jun 10 2012

%o (Magma) [(&+[MoebiusMu(k)^2*Floor(n/k)*Floor(1 + n/k): k in [1..n]])/2: n in [1..60]]; // _G. C. Greubel_, Nov 23 2018

%Y Cf. A001615, A063659.

%Y Cf. A082020.

%Y Cf. A175836 (partial products of the Dedekind psi function).

%K nonn

%O 1,2

%A _Jonathan Vos Post_, Feb 15 2010