A097988 a(n) = Sum_{d dividing n} tau(d)^3 = (Sum_{d dividing n} tau(d))^2.

1, 9, 9, 36, 9, 81, 9, 100, 36, 81, 9, 324, 9, 81, 81, 225, 9, 324, 9, 324, 81, 81, 9, 900, 36, 81, 100, 324, 9, 729, 9, 441, 81, 81, 81, 1296, 9, 81, 81, 900, 9, 729, 9, 324, 324, 81, 9, 2025, 36, 324, 81, 324, 9, 900, 81, 900, 81, 81, 9, 2916, 9, 81, 324

Offset: 1

Author

Lekraj Beedassy, Sep 07 2004

Keywords

• nonn
• mult
• easy

Comments

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.

3^A001221(n) always divides a(n) except if n > 1 and included in A000578. - Enrique Pérez Herrero, Jul 12 2010

References

• Tom M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 47.
• Jean-Marie De Koninck and Armel Mercier, 1001 Problèmes en Théorie Classique Des Nombres, Problem 562, pp. 75, 265; Ellipses Paris 2004.
• William J. LeVeque, Topics in Number Theory. Addison-Wesley, Reading, MA, 2 vols., 1956, Vol. 1, p. 85, Problem 2.
• William J. LeVeque, Fundamentals of Number Theory, Dover Publications Inc, 1977, p. 125.
• Joe Roberts, The Lure of Integers, MAA, 1992, Integer 3, pages 8-9.
• J. V. Uspensky and M. A. Heaslet, Elementary Number Theory, McGraw-Hill, NY, 1939, p. 84.

Links

• Seiichi Manyama, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Enrique Pérez Herrero)

Crossrefs

Cf. A000005, A000217, A000537, A007425.

Programs

Maple

with(numtheory); f:=proc(n) local t1; t1:=divisors(n); add(sigma[0](i), i in t1)^2; end;

Mathematica

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 *)
f[n_]:=Total[DivisorSigma[0,Divisors[n]]]^2;f/@Range[100] (* Ivan N. Ianakiev, Mar 05 2015 *)
a[n_] := DivisorSum[n, DivisorSigma[0, #]&]^2; Array[a, 100] (* Jean-François Alcover, Dec 02 2015 *)
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 *)

PARI

a(n)=sumdiv(n,d,numdiv(d))^2 \\ Charles R Greathouse IV, Jan 22 2013

PARI

a(n)=sumdiv(n, d, numdiv(d)^3); \\ Michel Marcus, Nov 21 2013

Formula

a(n) = (Sum_{d dividing n} tau(d))^2 = (A007425(n))^2.

Multiplicative with a(p^e) = ((e+1)*(e+2)/2)^2. - Amiram Eldar, Sep 20 2020

Dirichlet g.f.: zeta(s)^5 * Product_{p prime} (1 + 4/p^s + 1/p^(2*s)). - Amiram Eldar, Sep 14 2023

Extensions

More terms from Carl Najafi, Oct 19 2011

Entry revised by N. J. A. Sloane, May 22 2012