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A097988 a(n) = Sum_{d dividing n} tau(d)^3 = (Sum_{d dividing n} tau(d))^2. 5
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 (list; graph; refs; listen; history; text; internal format)
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

T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 47.

J.-M. De Koninck & A.Mercier, 1001 Problèmes en Théorie Classique Des Nombres, Problem 562, pp. 75, 265; Ellipses Paris 2004.

W. 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

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

Cf. A000005, A000217, A000537, A007425.

Sequence in context: A003874 A339735 A341835 * A103646 A246314 A341538

Adjacent sequences:  A097985 A097986 A097987 * A097989 A097990 A097991

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

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Last modified January 24 19:03 EST 2022. Contains 350565 sequences. (Running on oeis4.)