OFFSET
1,2
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)
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
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 *)
PROG
(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
CROSSREFS
KEYWORD
nonn,mult,easy
AUTHOR
Lekraj Beedassy, Sep 07 2004
EXTENSIONS
More terms from Carl Najafi, Oct 19 2011
Entry revised by N. J. A. Sloane, May 22 2012
STATUS
approved