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 A173290 Partial sums of A001615. 10
 1, 4, 8, 14, 20, 32, 40, 52, 64, 82, 94, 118, 132, 156, 180, 204, 222, 258, 278, 314, 346, 382, 406, 454, 484, 526, 562, 610, 640, 712, 744, 792, 840, 894, 942, 1014, 1052, 1112, 1168, 1240, 1282, 1378, 1422, 1494, 1566, 1638, 1686, 1782, 1838, 1928, 2000, 2084 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS a(n) is even for n >= 2. - Jianing Song, Nov 24 2018 REFERENCES 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 LINKS Enrique Pérez Herrero, Table of n, a(n) for n = 1..5000 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. FORMULA a(n) = Sum_{i=1..n} A001615(i) = Sum_{i=1..n} (n * Product_{p|n, p prime} (1 + 1/p)). a(n) = 15*n^2/(2*Pi^2) + O(n*log(n)). - Enrique Pérez Herrero, Jan 14 2012 a(n) = Sum_{i=1..n} A063659(i) * floor(n/i). - Enrique Pérez Herrero, Feb 23 2013 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 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 MAPLE 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 MATHEMATICA Table[Sum[DirichletConvolve[j, MoebiusMu[j]^2, j, k], {k, 1, n}], {n, 60}] (* G. C. Greubel, Nov 23 2018 *) psi[n_] := If[n==1, 1, n*Times@@(1 + 1/FactorInteger[n][[;; , 1]])]; Accumulate[Array[psi, 50]] (* Amiram Eldar, Nov 23 2018 *) PROG (PARI) S(n) = sum(k=1, sqrtint(n), moebius(k)*(n\(k*k))); \\ see: A013928 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 (Sage) def A173290(n) : return add(k*mul(1+1/p for p in prime_divisors(k)) for k in (1..n)) [A173290(n) for n in (1..52)] # Peter Luschny, Jun 10 2012 (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 CROSSREFS Cf. A001615, A063659. Cf. A082020. Cf. A175836 (partial products of the Dedekind psi function). Sequence in context: A055507 A343880 A121896 * A312686 A312687 A312688 Adjacent sequences: A173287 A173288 A173289 * A173291 A173292 A173293 KEYWORD nonn AUTHOR Jonathan Vos Post, Feb 15 2010 STATUS approved

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Last modified September 27 20:41 EDT 2023. Contains 365714 sequences. (Running on oeis4.)