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A173290 Partial sums of A001615. 4
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: A265284 A055507 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 November 28 06:23 EST 2020. Contains 338699 sequences. (Running on oeis4.)