A007875 Number of ways of writing n as p*q, with p <= q, gcd(p, q) = 1.

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, 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, 2, 4, 1, 2, 1, 2, 2, 2, 2, 4, 1, 2, 1, 2, 1, 4, 2, 2, 2, 2, 1, 4

Offset: 1

Author

Victor Ufnarovski

Keywords

• nonn
• nice
• easy

Comments

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

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

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

Links

• T. D. Noe, Table of n, a(n) for n = 1..1000
• Larry Bates and Peter Gibson, A geometry where everything is better than nice, arXiv:1603.06622 [math.DG], (21-March-2016); see page 2.
• Vaclav Kotesovec, Graph - the asymptotic ratio.

Crossrefs

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

Programs

Haskell

a007875 = length . filter (> 0) . a225817_row
-- Reinhard Zumkeller, Jul 30 2013, Aug 24 2011

Maple

A007875 := proc(n)
    if n = 1 then
        1;
    else
        2^(A001221(n)-1) ;
    end if;
end proc: # R. J. Mathar, May 28 2016

Mathematica

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 *)
a[n_] := EulerPhi[2^PrimeNu[n]]; Array[a, 105] (* Robert G. Wilson v, Apr 10 2012 *)
a[n_] := Sum[If[Mod[n, k] == 0, Re[Sqrt[MoebiusMu[k]]], 0], {k, 1, n}] (* Mats Granvik, Aug 10 2018 *)

PARI

a(n)=ceil((1<Charles R Greathouse IV, Nov 02 2011

Formula

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

a(n) = phi(2^omega(n)) = A000010(2^A001221(n)). - Enrique Pérez Herrero, Apr 10 2012

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

a(n) = Sum_{d|n} mu(d) * A018892(n/d). - Daniel Suteu, Jan 08 2021

Dirichlet g.f.: (zeta(s)^2/zeta(2*s) + 1)/2. - Amiram Eldar, Sep 09 2023