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 A048691 a(n) = d(n^2), where d(k) = A000005(k) is the number of divisors of k. 69

%I

%S 1,3,3,5,3,9,3,7,5,9,3,15,3,9,9,9,3,15,3,15,9,9,3,21,5,9,7,15,3,27,3,

%T 11,9,9,9,25,3,9,9,21,3,27,3,15,15,9,3,27,5,15,9,15,3,21,9,21,9,9,3,

%U 45,3,9,15,13,9,27,3,15,9,27,3,35,3,9,15,15,9,27,3,27

%N a(n) = d(n^2), where d(k) = A000005(k) is the number of divisors of k.

%C 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 Number of elements in the set {(x,y): x|n, y|n, gcd(x,y)=1}.

%C Number of elements in the set {(x,y): lcm(x,y)=n}.

%C 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-to-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_, May 31 2002

%C 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_, Jun 19 2005

%C a(n) = A055205(n) + A000005(n). - _Reinhard Zumkeller_, Dec 08 2009

%D A. M. Gleason et al., The William Lowell Putnam Mathematical Competitions, Problems & Solutions:1938-1960 Soln. to Prob. 1 1960 p. 516 MAA

%D R. Honsberger, More Mathematical Morsels, Morsel 43 pp. 232-3, DMA No. 10 MAA 1991

%D L. C. Larson, Problem-Solving Through Problems, Prob. 3.3.7, p. 102 Springer 1983

%D A. S. Posamentier & C. T. Salkind, Challenging Problems in Algebra, Prob. 9-9 pp. 143 Dover NY 1988

%D D. O. Shklarsky et al., The USSR Olympiad Problem Book, Soln. to Prob. 123 pp. 28;217-8 Dover NY

%D W. Sierpiński, Elementary Theory of Numbers, pp. 71-2 Elsevier, North Holland 1988

%D Charles W. Trigg, Mathematical Quickies, Question 194 pp. 53;168, Dover 1985

%H Enrique Pérez Herrero, <a href="/A048691/b048691.txt">Table of n, a(n) for n = 1..10000</a>

%H Ovidiu Bagdasar, <a href="/A048691/a048691.pdf">On Some Functions Involving the lcm and gcd of Integer Tuples</a>

%H Umberto Cerruti, <a href="/A146564/a146564.pdf">Percorsi tra i numeri</a> (in Italian), pages 3-4.

%H Daniele A. Gewurz and Francesca Merola, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL6/Gewurz/gewurz5.html">Sequences realized as Parker vectors of oligomorphic permutation groups</a>, J. Integer Seqs., Vol. 6, 2003.

%H J. Scholes, <a href="http://www.kalva.demon.co.uk/putnam/psoln/psol601.html">Problem A1 of 21st Putnam Competition 1960</a>

%H W. Sierpiński, <a href="http://matwbn.icm.edu.pl/kstresc.php?tom=42&amp;wyd=10">Elementary Theory of Numbers</a>, Warszawa 1964.

%F tau(n^2) = Sum_{d|n} mu(n/d)*tau(d)^2, where mu(n) = A008683(n), cf. A061391.

%F Multiplicative with a(p^e) = 2e+1. - _Vladeta Jovovic_, Jul 23 2001

%F Also a(n) = sum(d|n, tau(d)*moebius(n/d)^2). - _Benoit Cloitre_, Sep 08 2002

%F Dirichlet g.f. (zeta(s))^3/zeta(2s). - _R. J. Mathar_, Feb 11 2011

%F a(n) = Sum_{d|n} 2^omega(d). - _Enrique Pérez Herrero_, Apr 14 2012

%F a(n) = A000005(A000290(n)). - _Omar E. Pol_, Oct 24 2013

%F G.f.: Sum_{k>=1} 2^omega(k)*x^k/(1 - x^k). - _Ilya Gutkovskiy_, Mar 10 2018

%t A048691[n_]:=DivisorSigma[0,n^2] (* _Enrique Pérez Herrero_, May 30 2010 *)

%t DivisorSigma[0,Range[80]^2] (* _Harvey P. Dale_, Apr 08 2015 *)

%o (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_, Dec 30 2007

%o (MuPAD) numlib::tau (n^2)\$ n=1..90 // _Zerinvary Lajos_, May 13 2008

%o a048691 = product . map (a005408 . fromIntegral) . a124010_row

%o -- _Reinhard Zumkeller_, Jul 12 2012

%Y Cf. A000005, A035116, A048785, A061391, A018805, A002088, A015614, A018892, A063647, A182139.

%Y Partial sums give A061503.

%Y For similar LCM sequences, see A070919, A070920, A070921.

%Y Cf. A005408, A124010.

%K nonn,mult

%O 1,2

%A _David Johnson-Davies_