A153818 a(n) = Sum_{k=1..n} floor(n^2/k^2).
1, 5, 12, 22, 35, 53, 72, 96, 123, 153, 184, 222, 260, 304, 351, 402, 453, 510, 568, 633, 697, 765, 839, 916, 994, 1077, 1164, 1252, 1342, 1443, 1535, 1641, 1747, 1856, 1969, 2083, 2200, 2321, 2447, 2579, 2705, 2844, 2979, 3123, 3269, 3417, 3570, 3726, 3881
Offset: 1
Examples
a(4)=22 because floor(16/1) + floor(16/4) + floor(16/9) + floor(16,16) = 16 + 4 + 1 + 1 = 22. [_Emeric Deutsch_, Jan 13 2009]
Links
- Seiichi Manyama, Table of n, a(n) for n = 1..10000
- Benoit Cloitre, Plot of (a(n)-zeta(2)*n^2-zeta(1/2)*n)/(n^0.5/log(n))
Programs
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Maple
a := proc (n) options operator, arrow: sum(floor(n^2/k^2), k = 1 .. n) end proc: seq(a(n), n = 1 .. 50); # Emeric Deutsch, Jan 13 2009
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PARI
a(n)=sum(k=1,n,n^2\k^2) \\ Benoit Cloitre, Jan 22 2013
Formula
From Benoit Cloitre, Jan 22 2013: (Start)
Asymptotic formula: a(n) = zeta(2)*n^2 + zeta(1/2)*n + O(n^(1/2)).
Conjecture: a(n) = zeta(2)*n^2 + zeta(1/2)*n + O(n^0.5/log(n)) (see link). (End)
Extensions
Definition edited by Emeric Deutsch, Jan 13 2009
Extended by Emeric Deutsch, Jan 13 2009
Comments