A069087 -id:A069087 - cp's OEIS frontend

A069891 a(n) = Sum_{k=1..n} A007913(k), the squarefree part of k.

0, 1, 3, 6, 7, 12, 18, 25, 27, 28, 38, 49, 52, 65, 79, 94, 95, 112, 114, 133, 138, 159, 181, 204, 210, 211, 237, 240, 247, 276, 306, 337, 339, 372, 406, 441, 442, 479, 517, 556, 566, 607, 649, 692, 703, 708, 754, 801, 804, 805, 807, 858, 871, 924, 930, 985, 999

Offset: 0

Dean Hickerson, Apr 09 2002

nonn

Sum_{k=1..n, k squarefree} (1/k) = Sum{k=1..n} (mu(k)^2/k) = (1/zeta(2))*(log(n) + gamma - 2*zeta'(2)/zeta(2)) + O(1/sqrt(n)). (Suryanarayana)

D. Suryanarayana, Asymptotic formula for sum_{n <= x} mu^{2}(n)/n, Indian J. Math. 9 (1967/1968) 543-545.

Amiram Eldar, Table of n, a(n) for n = 0..10000
Rafael Jakimczuk, Two Topics in Number Theory: Sum of Divisors of the Primorial and Sum of Squarefree Parts, International Mathematical Forum, Vol. 12, 2017, no. 7, pp. 331-338.

Cf. A007913, A046970, A069087.

[0] cat [&+[Squarefree(k):k in [1..n]]:n in [1..60]]; // Marius A. Burtea, Dec 19 2019

a[n_] := Sum[If[d == 1, 1, Times@@(1-#1[[1]]^2&) /@ FactorInteger[d]] * Binomial[Floor[n/d^2]+1, 2], {d, 1, Floor[Sqrt[n]]}]; Array[a, 100, 0] (* corrected by Amiram Eldar, Apr 02 2020 *)

a(n) = sum(k=1, n, core(k)); \\ Michel Marcus, Dec 19 2019

a(n) = Sum_{d=1..floor(sqrt(n))} f(d)*binomial(floor(n/d^2)+1, 2) where f(d)=A046970(d) is the product of 1-p^2 over all prime divisors p of d.

a(n) is asymptotic to r*n^2, where r = Pi^2/30 = 0.3289868...

A100070 Number a(n) of forests with two components in the complete bipartite graph K_{n,n}.

Original entry on oeis.org

6, 117, 5632, 515625, 77262336, 17230990189, 5360119185408, 2219048868131217, 1180000000000000000, 783948341202404638821, 636404158746280870281216, 619884903445287035295372217, 713552333492738487958741450752

Offset: 2

Woong Kook (andrewk(AT)math.uri.edu), Nov 02 2004

nonn

This sequence (a(n)) appears to dominate the sequence (n^{2n-2}) of the number of spanning trees in K_{n,n} for n>1. This shows that the sequence of independent set numbers for the cycle matroid of K_{n,n} is not monotone increasing unlike the complete graph K_{n}.

			a(2)=6 because K_{2,2} is C_{4} the cycle of length 4 and there are 6 forests with two components in C_{4}.

G. C. Greubel, Table of n, a(n) for n = 2..214
N. Eaton, W. Kook and L. Thoma, Monotonicity for complete graphs, preprint, 2004.

Cf. A000272, A069087, A083483.

a[n_]:=Sum[Binomial[n, x]*Binomial[n, y]*x^(y-1)*y^(x-1)*(n-x)^(n-y-1)*(n-y)^(n-x-1), {x, 1, n-1}, {y, 1, n-1}]/2 + (2*(n^2-n)^(n-1)); Table[a[n], {n, 2, 10}] (* This will generate a(n) from n=2 to 10. *)

a(n) = 2*(n^2 - n)^(n-1) + (1/2)*Sum_{x=1..(n-1)} Sum_{y=1..(n-1)} b(n, x, y), where b(n, x, y) = binomial(n,x)*binomial(n,y)*x^(y-1)*y^(x-1)*(n-x)^(n-y-1)*(n-y)^(n-x-1).

cp's OEIS Frontend

A069891 a(n) = Sum_{k=1..n} A007913(k), the squarefree part of k.

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