A007875 - OEIS

%I #78 Sep 09 2023 06:49:26

%S 1,1,1,1,1,2,1,1,1,2,1,2,1,2,2,1,1,2,1,2,2,2,1,2,1,2,1,2,1,4,1,1,2,2,

%T 2,2,1,2,2,2,1,4,1,2,2,2,1,2,1,2,2,2,1,2,2,2,2,2,1,4,1,2,2,1,2,4,1,2,

%U 2,4,1,2,1,2,2,2,2,4,1,2,1,2,1,4,2,2,2,2,1,4

%N Number of ways of writing n as p*q, with p <= q, gcd(p, q) = 1.

%C a(n), n >= 2, is the number of divisor products in the numerator as well as denominator of the unique representation of n in terms of divisor products. See the W. Lang link under A007955, where a(n)=l(n) in Table 1. - _Wolfdieter Lang_, Feb 08 2011

%C Record values are the binary powers, occurring at primorial positions except at 2: a(A002110(0))=A000079(0), a(A002110(n+1))=A000079(n) for n > 0. - _Reinhard Zumkeller_, Aug 24 2011

%C For n > 1: a(n) = (A000005(n) - A048105(n)) / 2; number of ones in row n of triangle in A225817. - _Reinhard Zumkeller_, Jul 30 2013

%H T. D. Noe, <a href="/A007875/b007875.txt">Table of n, a(n) for n = 1..1000</a>

%H Larry Bates and Peter Gibson, <a href="http://arxiv.org/abs/1603.06622">A geometry where everything is better than nice</a>, arXiv:1603.06622 [math.DG], (21-March-2016); see page 2.

%H Vaclav Kotesovec, <a href="/A007875/a007875.jpg">Graph - the asymptotic ratio</a>.

%F a(n) = (1/2)*Sum_{ d divides n } abs(mu(d)) = 2^(A001221(n)-1) = A034444(n)/2, n > 1. - _Vladeta Jovovic_, Jan 25 2002

%F a(n) = phi(2^omega(n)) = A000010(2^A001221(n)). - _Enrique Pérez Herrero_, Apr 10 2012

%F Sum_{k=1..n} a(k) ~ 3*n*((log(n) + (2*gamma - 1))/ Pi^2 - 12*(zeta'(2)/Pi^4)), where gamma is the Euler-Mascheroni constant A001620. Equivalently, Sum_{k=1..n} a(k) ~ 3*n*(log(n) + 24*log(A) - 1 - 2*log(2*Pi)) / Pi^2, where A is the Glaisher-Kinkelin constant A074962. - _Vaclav Kotesovec_, Jan 30 2019

%F a(n) = Sum_{d|n} mu(d) * A018892(n/d). - _Daniel Suteu_, Jan 08 2021

%F Dirichlet g.f.: (zeta(s)^2/zeta(2*s) + 1)/2. - _Amiram Eldar_, Sep 09 2023

%p A007875 := proc(n)

%p     if n = 1 then

%p         1;

%p     else

%p         2^(A001221(n)-1) ;

%p     end if;

%p end proc: # _R. J. Mathar_, May 28 2016

%t a[n_] := With[{r = Reduce[1 <= p <= q <= n && n == p*q && GCD[p, q] == 1, {p, q}, Integers]}, If[Head[r] === And, 1, Length[r]]]; Table[a[n], {n, 1, 90}] (* _Jean-François Alcover_, Nov 02 2011 *)

%t a[n_] := EulerPhi[2^PrimeNu[n]]; Array[a, 105] (* _Robert G. Wilson v_, Apr 10 2012 *)

%t a[n_] := Sum[If[Mod[n, k] == 0, Re[Sqrt[MoebiusMu[k]]], 0], {k, 1, n}] (* _Mats Granvik_, Aug 10 2018 *)

%o (Haskell)

%o a007875 = length . filter (> 0) . a225817_row

%o -- _Reinhard Zumkeller_, Jul 30 2013, Aug 24 2011

%o (PARI) a(n)=ceil((1<<omega(n))/2) \\ _Charles R Greathouse IV_, Nov 02 2011

%Y Cf. A000005, A000010, A000079, A001221, A001620, A002110, A007955, A034444, A048105, A074962.

%K nonn,nice,easy

%O 1,6

%A Victor Ufnarovski