%I A048691
%S A048691 1,3,3,5,3,9,3,7,5,9,3,15,3,9,9,9,3,15,3,15,9,9,3,21,5,9,7,15,3,
%T A048691 27,3,11,9,9,9,25,3,9,9,21,3,27,3,15,15,9,3,27,5,15,9,15,3,21,9,
%U A048691 21,9,9,3,45,3,9,15,13,9,27,3,15,9,27,3,35,3,9,15,15,9,27,3,27
%N A048691 tau(n^2), where tau = A000005.
%C A048691 A048691 is the inverse Moebius transform of A034444: Sum_{d|n} 2^omega(d), 
               where omega(n) = A001221(n) is the number of distinct primes dividing 
               n.
%C A048691 Number of elements in the set {(x,y): x|n, y|n, gcd(x,y)=1}.
%C A048691 Number of elements in the set {(x,y): lcm(x,y)=n}.
%C A048691 Also gives total number of positive integral solutions (x,y), order being 
               taken into account, to the optical or parallel resistor equation 
               1/x + 1/y = 1/n. Indeed, writing the latter as X*Y=N, with X=x-n, 
               Y=y-n, N=n^2, the one-one correspondence between solutions (X, Y) 
               and (x, y) is obvious, so that clearly, the solution pairs (x, y) 
               are tau(N)=tau(n^2) in number. - Lekraj Beedassy (blekraj(AT)yahoo.com), 
               May 31 2002
%C A048691 Number of ordered pairs of positive integers (a,c) such that n^2 - ac 
               = 0. Therefore number of quadratic equations of the form ax^2 + 2nx 
               + c = 0 where a,n,c are positive integers and each equation has two 
               equal (rational) roots, -n/a. (If a and c are positive integers, 
               but, instead, the coefficient of x is odd, it is impossible for the 
               equation to have equal roots.) - Rick L. Shepherd (rshepherd2(AT)hotmail.com), 
               Jun 19 2005
%D A048691 A. M. Gleason et al., The William Lowell Putnam Mathematical Competitions, 
               Problems & Solutions:1938-1960 Soln. to Prob. 1 1960 p. 516 MAA
%D A048691 R. Honsberger, More Mathematical Morsels, Morsel 43 pp. 232-3, DMA No. 
               10 MAA 1991
%D A048691 L. C. Larson, Problem-Solving Through Problems, Prob. 3.3.7, p. 102 Springer 
               1983
%D A048691 A. S. Posamentier & C. T. Salkind, Challenging Problems in Algebra, Prob. 
               9-9 pp. 143 Dover NY 1988
%D A048691 D. O. Shklarsky et al., The USSR Olympiad Problem Book, Soln. to Prob. 
               123 pp. 28;217-8 Dover NY
%D A048691 W. Sierpinski, Elementary Theory of Numbers, pp. 71-2 Elsevier, North 
               Holland 1988
%D A048691 Charles W. Trigg, Mathematical Quickies, Question 194 pp. 53;168, Dover 
               1985
%H A048691 Daniele A. Gewurz and Francesca Merola, <a href="http://www.cs.uwaterloo.ca/
               journals/JIS/index.html">Sequences realized as Parker vectors ...</
               a>, J. Integer Seqs., Vol. 6, 2003.
%H A048691 J. Scholes, <a href="http://www.kalva.demon.co.uk/putnam/psoln/psol601.html">
               Problem A1 of 21-st Putnam Competition 1960</a>
%H A048691 W. Sierpi\'{n}ski, <a href="http://matwbn.icm.edu.pl/kstresc.php?tom=42&wyd=10">
               Elementary Theory of Numbers</a>, Warszawa 1964.
%F A048691 tau(n^2) = Sum_{d|n} mu(n/d)*tau(d)^2, where mu(n) = A008683(n), cf. 
               A061391.
%F A048691 Multiplicative with a(p^e) = 2e+1. - Vladeta Jovovic (vladeta(AT)eunet.rs), 
               Jul 23 2001
%F A048691 Also a(n) = sum(d|n, tau(d)*moebius(n/d)^2 ) - Benoit Cloitre (benoit7848c(AT)orange.fr), 
               Sep 08 2002
%o A048691 (PARI) A048691(n)=prod(i=1,#n=factor(n)[,2],n[i]*2+1) /* or, of course, 
               a(n)=numdiv(n^2) */ - M. F. Hasler (Maximilian.Hasler(AT)gmail.com), 
               Dec 30 2007
%o A048691 (MuPad)numlib::tau (n^2)$ n=1..90 - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), 
               May 13 2008
%Y A048691 Cf. A035116, A061391, A000005, A018805, A002088, A015614, A018892, A063647.
%Y A048691 Partial sums give A061503.
%Y A048691 For similar LCM sequences, see A070919-A070921.
%Y A048691 Sequence in context: A096866 A015909 A029620 this_sequence A071053 A094439 
               A122037
%Y A048691 Adjacent sequences: A048688 A048689 A048690 this_sequence A048692 A048693 
               A048694
%K A048691 nonn,mult
%O A048691 1,2
%A A048691 David Johnson-Davies (David(AT)interface.co.uk)
%E A048691 Additional comments from Vladeta Jovovic (vladeta(AT)eunet.rs), Apr 29 
               2001