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A007875 Number of ways of writing n as pq, with p<=q, (p,q)=1. 12
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,6

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, Peter Gibson, A geometry where everything is better than nice, arXiv:1603.06622 [math.DG], (21-March-2016); see page 2

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

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 *)

f[n_] := EulerPhi[2^PrimeNu[n]]; Array[f, 105] (* Robert G. Wilson v, Apr 10 2012 *)

PROG

(Haskell)

a007875 = length . filter (> 0) . a225817_row

-- Reinhard Zumkeller, Jul 30 2013, Aug 24 2011.

(PARI) a(n)=ceil((1<<omega(n))/2) \\ Charles R Greathouse IV, Nov 02 2011

CROSSREFS

Sequence in context: A001221 A064372 A096825 * A259936 A050320 A121382

Adjacent sequences:  A007872 A007873 A007874 * A007876 A007877 A007878

KEYWORD

nonn,nice,easy

AUTHOR

Victor Ufnarovski

STATUS

approved

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Last modified August 21 00:45 EDT 2017. Contains 290855 sequences.