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A061201 Partial sums of A007425: (tau<=)_3(n). 17
1, 4, 7, 13, 16, 25, 28, 38, 44, 53, 56, 74, 77, 86, 95, 110, 113, 131, 134, 152, 161, 170, 173, 203, 209, 218, 228, 246, 249, 276, 279, 300, 309, 318, 327, 363, 366, 375, 384, 414, 417, 444, 447, 465, 483, 492, 495, 540, 546, 564, 573, 591, 594, 624, 633, 663 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

(tau<=)_k(n) = |{(x_1,x_2,...,x_k): x_1*x_2*...*x_k<=n}|, i.e., tau<=_k(n) is number of solutions to x_1*x_2*...*x_k<=n, x_i > 0.

A061201(n) is the number of 4-tuples (w,x,y,z) having all terms in {1,...,n} and w=x*y*z; see A211795 for a list of related counting sequences. - Clark Kimberling, Apr 28 2012

REFERENCES

M. N. Huxley, Area, Lattice Points and Exponential Sums, Oxford, 1996; p. 239.

LINKS

Vaclav Kotesovec, Table of n, a(n) for n = 1..100000 (terms 1..1000 from Harry J. Smith)

Vaclav Kotesovec, Graph - The asymptotic ratio (1000000 terms)

FORMULA

(tau<=)_k(n) = Sum_{i=1..n} tau_k(i).

a(n) = n * ( log(n)^2/2 + (3*g-1)*log(n) + 3*g^2-3*g-3*g1+1 ) + O(sqrt(n)), where g is the Euler-Mascheroni number ~ 0.57721... (see A001620), and g1 is the first Stieltjes constant ~ -0.072816 (see A082633). The determination of the precise size of the error term is an unsolved problem - see references. - Andrew Lelechenko, Apr 15 2011 [corrected by Vaclav Kotesovec, Sep 09 2018]

a(n) = Sum_{k=1..n} A000005(k)*floor(n/k). - Benoit Cloitre, Apr 19 2007

To compute a(n) for huge n (see A180365) in sublinear use a(n) = 3*Sum_{i=1..n3} A006218(n/i) - Sum_{j=1..n3} floor(n/(i*j)) + n3^3, where n3 = floor(n^(1/3)). - Andrew Lelechenko, Apr 15 2011

a(n) = Sum_{k=1..n} Sum_{i=1..n} floor(n/(i*k)). - Wesley Ivan Hurt, Sep 14 2017

MATHEMATICA

a[n_] := Sum[ DivisorSigma[0, k]*Floor[n/k], {k, 1, n}]; Table[a[n], {n, 1, 56}] (* Jean-Fran├žois Alcover, Sep 20 2011, after Benoit Cloitre *)

(* Asymptotics: *) n*(Log[n]^2/2 + (3*EulerGamma - 1)*Log[n] + 3*EulerGamma^2 - 3*EulerGamma - 3*StieltjesGamma[1] + 1) (* Vaclav Kotesovec, Sep 09 2018 *)

Accumulate[a[n_]:=DivisorSum[n, DivisorSigma[0, #]&]; Array[a, 60]] (* Vincenzo Librandi, Jan 12 2020 *)

PROG

(PARI) a(n)=sum(k=1, n, numdiv(k)*floor(n/k)) \\ Benoit Cloitre, Apr 19 2007

(PARI) { for (n=1, 1000, write("b061201.txt", n, " ", sum(k=1, n, numdiv(k)*(n\k))) ) } \\ Harry J. Smith, Jul 18 2009

(MAGMA) [&+[NumberOfDivisors(k)*Floor(n/k): k in [1..n]]: n in [1..56]];  // Bruno Berselli, Apr 13 2011

CROSSREFS

Cf. tau_2(n): A000005, tau_3(n): A007425, tau_4(n): A007426, tau_5(n): A061200, tau_6(n): A034695, (unordered) 2-factorizations of n: A038548, (unordered) 3-factorizations of n: A034836, A001055, (tau<=)_2(n): A006218, (tau<=)_4(n): A061202, (tau<=)_5(n): A061203, (tau<=)_6(n): A061204.

Sequence in context: A310817 A309794 A048297 * A219907 A310818 A310819

Adjacent sequences:  A061198 A061199 A061200 * A061202 A061203 A061204

KEYWORD

easy,nonn

AUTHOR

Vladeta Jovovic, Apr 21 2001

STATUS

approved

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Last modified August 3 18:51 EDT 2020. Contains 336201 sequences. (Running on oeis4.)