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A222548
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a(n) = Sum_{k=1..n} floor(n/k)^2.
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19
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1, 5, 11, 22, 32, 52, 66, 92, 115, 147, 169, 219, 245, 289, 333, 390, 424, 496, 534, 612, 672, 740, 786, 898, 957, 1037, 1113, 1219, 1277, 1413, 1475, 1595, 1687, 1791, 1883, 2056, 2130, 2246, 2354, 2526, 2608, 2792, 2878, 3040, 3190, 3330, 3424, 3662, 3773
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OFFSET
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1,2
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COMMENTS
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a(n) is the number of common divisors of integers 1<=i,j<=n over all ordered pairs (i,j). - Geoffrey Critzer, Jan 15 2015
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REFERENCES
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J. V. Uspensky and M. A. Heaslet, Elementary Number Theory, McGraw-Hill, NY, 1939, p. 98.
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LINKS
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FORMULA
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a(n) = zeta(2)*n^2 + O(n log n).
G.f.: (1/(1 - x)) * Sum_{k>=1} (2*k - 1) * x^k/(1 - x^k). - Ilya Gutkovskiy, Jul 16 2019
a(n) = Sum_{d=1..n} (2*d-1)*floor(n/d). [Uspensky and Heaslet] - Michael Somos, Feb 16 2020
a(n) = Sum_{k=1..n} Sum_{d|k} floor(n/d). - Ridouane Oudra, Jul 16 2020
a(n) = Sum_{i=1..n} Sum_{j=1..n} tau(gcd(i,j)). - Ridouane Oudra, Nov 23 2021
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MATHEMATICA
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Table[Sum[Floor[n/k]^2, {k, n}], {n, 50}] (* T. D. Noe, Feb 26 2013 *)
Table[nn = n; Total[Level[Table[Table[DivisorSigma[0, GCD[i, j]], {i, 1, nn}], {j, 1, nn}], {2}]], {n, 1, 49}] (* Geoffrey Critzer, Jan 15 2015 *)
Table[Sum[2*DivisorSigma[1, k] - DivisorSigma[0, k], {k, 1, n}], {n, 1, 50}] (* Vaclav Kotesovec, Sep 02 2018 *)
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PROG
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(PARI) a(n)=sum(k=1, n, (n\k)^2)
(Magma) [&+[Floor(n/k)^2:k in [1..n] ]: n in [1..40]]; // Marius A. Burtea, Jul 16 2019
(Python)
from math import isqrt
def A222548(n): return -(s:=isqrt(n))**3 + sum((q:=n//k)*((k<<1)+q-1) for k in range(1, s+1)) # Chai Wah Wu, Oct 21 2023
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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