A098235 Number of ways to write n as a sum of two ordered positive squarefree numbers.

0, 1, 2, 3, 2, 3, 4, 6, 4, 3, 4, 7, 6, 5, 6, 10, 8, 8, 6, 11, 8, 9, 8, 14, 10, 9, 10, 13, 10, 9, 10, 16, 12, 13, 12, 22, 14, 13, 14, 22, 16, 15, 18, 25, 20, 15, 16, 26, 20, 16, 14, 27, 20, 20, 14, 26, 20, 21, 18, 29, 22, 21, 22, 30, 22, 21, 22, 35, 24, 25, 22, 42, 26, 27, 26, 39

Offset: 1

Author

Ralf Stephan, Aug 31 2004

Keywords

• nonn

Comments

a(n) ~ n * Prod[p prime, (1-2/p^2) * Prod[p^2|n, (p^2-1)/(p^2-2)]].

Examples

			a(12)=7 because 12=1+11=2+10=5+7=6+6=7+5=10+2=11+1.
		

Links

• T. D. Noe, Table of n, a(n) for n = 1..10000
• P. Pollack, Analytic and Combinatorial Number Theory, Course Notes, p. 122, 202. [?Broken link]
• P. Pollack, Analytic and Combinatorial Number Theory, Course Notes, p. 122, 202.

Crossrefs

Cf. A005117, A098236, A071068.

Programs

Mathematica

Join[{0}, Table[Sum[(MoebiusMu[k]*MoebiusMu[n - k + 1])^2, {k, 1, n}], {n, 1, 50}]] (* G. C. Greubel, Dec 28 2016 *)

PARI

a(n) = sum(k=1, n-1, (moebius(k)*moebius(n-k))^2) \\ Indranil Ghosh, Mar 10 2017

PARI

a(n)=my(s); forsquarefree(k=1, n-1, s+=issquarefree(n-k)); s \\ Charles R Greathouse IV, Jan 08 2018

Formula

a(n) = Sum_{k=1..n-1} (mu(k)*mu(n-k))^2. - Benoit Cloitre, Sep 24 2006

a(n) = Sum_{k=1..n-1} ( A008966(k)*A008966(n-k) ). - Reinhard Zumkeller, Nov 04 2009

G.f.: ( Sum_{k>=1} mu(k)^2*x^k )^2, where mu(k) is the Moebius function (A008683). - Ilya Gutkovskiy, Dec 28 2016