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%I #62 Sep 14 2023 02:30:59
%S 1,9,9,36,9,81,9,100,36,81,9,324,9,81,81,225,9,324,9,324,81,81,9,900,
%T 36,81,100,324,9,729,9,441,81,81,81,1296,9,81,81,900,9,729,9,324,324,
%U 81,9,2025,36,324,81,324,9,900,81,900,81,81,9,2916,9,81,324
%N a(n) = Sum_{d dividing n} tau(d)^3 = (Sum_{d dividing n} tau(d))^2.
%C When n = p^e is a prime power, we have the corollary a(n) = Sum_{r=1..e+1} r^3 = (Sum_{r=1..e+1} r)^2, i.e. A000537(n) = (A000217(n))^2.
%C 3^A001221(n) always divides a(n) except if n > 1 and included in A000578. - _Enrique Pérez Herrero_, Jul 12 2010
%D Tom M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 47.
%D Jean-Marie De Koninck and Armel Mercier, 1001 Problèmes en Théorie Classique Des Nombres, Problem 562, pp. 75, 265; Ellipses Paris 2004.
%D William J. LeVeque, Topics in Number Theory. Addison-Wesley, Reading, MA, 2 vols., 1956, Vol. 1, p. 85, Problem 2.
%D William J. LeVeque, Fundamentals of Number Theory, Dover Publications Inc, 1977, p. 125.
%D Joe Roberts, The Lure of Integers, MAA, 1992, Integer 3, pages 8-9.
%D J. V. Uspensky and M. A. Heaslet, Elementary Number Theory, McGraw-Hill, NY, 1939, p. 84.
%H Seiichi Manyama, <a href="/A097988/b097988.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..1000 from Enrique Pérez Herrero)
%F a(n) = (Sum_{d dividing n} (tau(d))^2 = (A007425(n))^2.
%F Multiplicative with a(p^e) = ((e+1)*(e+2)/2)^2. - _Amiram Eldar_, Sep 20 2020
%F Dirichlet g.f.: zeta(s)^5 * Product_{p prime} (1 + 4/p^s + 1/p^(2*s)). - _Amiram Eldar_, Sep 14 2023
%p with(numtheory); f:=proc(n) local t1; t1:=divisors(n); add(sigma[0](i), i in t1)^2; end;
%t tau[1,n_Integer] := 1; SetAttributes[tau, Listable]; tau[k_Integer,n_Integer] := Plus@@(tau[k-1,Divisors[n]]); A097988[n_] := tau[3,n]^2; Table[A097988[n], {n, 100}] (* _Enrique Pérez Herrero_, Jul 12 2010 *)
%t f[n_]:=Total[DivisorSigma[0,Divisors[n]]]^2;f/@Range[100] (* _Ivan N. Ianakiev_, Mar 05 2015 *)
%t a[n_] := DivisorSum[n, DivisorSigma[0, #]&]^2; Array[a, 100] (* _Jean-François Alcover_, Dec 02 2015 *)
%t f[p_, e_] := ((e+1)*(e+2)/2)^2; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* _Amiram Eldar_, Sep 20 2020 *)
%o (PARI) a(n)=sumdiv(n,d,numdiv(d))^2 \\ _Charles R Greathouse IV_, Jan 22 2013
%o (PARI) a(n)=sumdiv(n, d, numdiv(d)^3); \\ _Michel Marcus_, Nov 21 2013
%Y Cf. A000005, A000217, A000537, A007425.
%K nonn,mult,easy
%O 1,2
%A _Lekraj Beedassy_, Sep 07 2004
%E More terms from _Carl Najafi_, Oct 19 2011
%E Entry revised by _N. J. A. Sloane_, May 22 2012
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