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A094820 Partial sums of A038548. 13

%I

%S 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,

%T 48,50,53,54,58,59,62,64,66,68,73,74,76,78,82,83,87,88,91,94,96,97,

%U 102,104,107,109,112,113,117,119,123,125,127,128,134,135,137,140,144,146,150

%N Partial sums of A038548.

%C a(n) = number of pairs (c,d) of integers such that 0 < c <= d and c*d <= n. - _Clark Kimberling_, Jun 18 2011

%C 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

%H Peter Kagey, <a href="/A094820/b094820.txt">Table of n, a(n) for n = 1..10000</a>

%H Vaclav Kotesovec, <a href="/A094820/a094820.jpg">Graph - the asymptotic ratio (100000 terms)</a>

%F G.f.: (1/(1 - x))*Sum_{k>=1} x^(k^2)/(1 - x^k). - _Ilya Gutkovskiy_, Apr 13 2017

%F 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

%p ListTools:-PartialSums([seq(ceil(numtheory:-tau(n)/2), n=1..100)]); # _Robert Israel_, Feb 24 2016

%t f[n_, k_] := Floor[n/k] - Floor[(n - 1)/k]

%t g[n_, k_] := If[k^2 <= n, f[n, k], 0]

%t Table[Sum[f[n, k], {k, 1, n}], {n, 1, 100}] (* A000005 *)

%t t = Table[Sum[g[n, k], {k, 1, n}], {n, 1, 100}]

%t (* A038548 *)

%t a[n_] := Sum[t[[i]], {i, 1, n}]

%t Table[a[n], {n, 1, 100}] (* A094820 *)

%t (* from Clark Kimberling, Jun 18 2011 *)

%t Table[Sum[Boole[d <= Sqrt[n]], {d, Divisors[n]}], {n, 1, 66}] // Accumulate (* _Jean-Fran├žois Alcover_, Dec 13 2012 *)

%o (Ruby)

%o def a(n)

%o (1..Math.sqrt(n)).inject(0) { |accum, i| accum + 1 + (n/i).to_i - i }

%o end # _Peter Kagey_, Feb 24 2016

%o (PARI) a(n) = sum(k=1, n, ceil(numdiv(k)/2)); \\ _Michel Marcus_, Feb 24 2016

%Y Cf. A091626, A038548.

%K easy,nonn

%O 1,2

%A _Vladeta Jovovic_, Jun 12 2004

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Last modified July 24 04:25 EDT 2021. Contains 346273 sequences. (Running on oeis4.)