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

1,3

Comments

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

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.

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

Example

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

Mathematica

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

Prog

(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

Crossrefs

Cf. A005117, A098236, A071068.

Sequence in context: A255395 A175266 A320053 * A342847 A345873 A114868

Adjacent sequences: A098232 A098233 A098234 * A098236 A098237 A098238

Keyword

nonn

Author

Ralf Stephan, Aug 31 2004

Status

approved