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A094820
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Partial sums of A038548.
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16
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1, 2, 3, 5, 6, 8, 9, 11, 13, 15, 16, 19, 20, 22, 24, 27, 28, 31, 32, 35, 37, 39, 40, 44, 46, 48, 50, 53, 54, 58, 59, 62, 64, 66, 68, 73, 74, 76, 78, 82, 83, 87, 88, 91, 94, 96, 97, 102, 104, 107, 109, 112, 113, 117, 119, 123, 125, 127, 128, 134, 135, 137, 140, 144, 146, 150
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OFFSET
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1,2
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COMMENTS
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a(n) = number of pairs (c,d) of integers such that 0 < c <= d and c*d <= n. - Clark Kimberling, Jun 18 2011
Equivalently, the number of representations of n in the form x + y*z, where x, y, and z are positive integers and y <= z. - John W. Layman, Feb 21 2012
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LINKS
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FORMULA
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G.f.: (1/(1 - x))*Sum_{k>=1} x^(k^2)/(1 - x^k). - Ilya Gutkovskiy, Apr 13 2017
a(n) ~ (log(n) + 2*gamma - 1)*n/2 + sqrt(n)/2, where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Aug 19 2019
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MAPLE
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ListTools:-PartialSums([seq(ceil(numtheory:-tau(n)/2), n=1..100)]); # Robert Israel, Feb 24 2016
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MATHEMATICA
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f[n_, k_] := Floor[n/k] - Floor[(n - 1)/k]
g[n_, k_] := If[k^2 <= n, f[n, k], 0]
Table[Sum[f[n, k], {k, 1, n}], {n, 1, 100}] (* A000005 *)
t = Table[Sum[g[n, k], {k, 1, n}], {n, 1, 100}]
a[n_] := Sum[t[[i]], {i, 1, n}]
Table[a[n], {n, 1, 100}] (* A094820 *)
(* from Clark Kimberling, Jun 18 2011 *)
Table[Sum[Boole[d <= Sqrt[n]], {d, Divisors[n]}], {n, 1, 66}] // Accumulate (* Jean-François Alcover, Dec 13 2012 *)
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PROG
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(Ruby)
def a(n)
(1..Math.sqrt(n)).inject(0) { |accum, i| accum + 1 + (n/i).to_i - i }
(PARI) a(n) = sum(k=1, n, ceil(numdiv(k)/2)); \\ Michel Marcus, Feb 24 2016
(Python)
from math import isqrt
def A094820(n): return ((s:=isqrt(n))*(1-s)>>1)+sum(n//k for k in range(1, s+1)) # Chai Wah Wu, Oct 23 2023
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CROSSREFS
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KEYWORD
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easy,nonn
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AUTHOR
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STATUS
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approved
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