A000082 a(n) = n^2*Product_{p|n} (1 + 1/p).

1, 6, 12, 24, 30, 72, 56, 96, 108, 180, 132, 288, 182, 336, 360, 384, 306, 648, 380, 720, 672, 792, 552, 1152, 750, 1092, 972, 1344, 870, 2160, 992, 1536, 1584, 1836, 1680, 2592, 1406, 2280, 2184, 2880, 1722, 4032, 1892, 3168, 3240, 3312, 2256

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• easy
• nice
• mult

Comments

For n > 1: A006530(a(n)) = A076566(n-1). - Reinhard Zumkeller, Oct 03 2012

A strong divisibility sequence, that is, gcd(a(n), a(m)) = a(gcd(n, m)) for all positive integers n and m. - Michael Somos, Jan 01 2017

References

• B. Schoeneberg, Elliptic Modular Functions, Springer-Verlag, NY, 1974, p. 79.

Links

• T. D. Noe, Table of n, a(n) for n = 1..1000
• Index to divisibility sequences

Crossrefs

Cf. A001615, A027748, A033196, A124010, A335762.

Programs

Haskell

a000082 n = product $ zipWith (\p e -> p ^ (2*e - 1) * (p + 1))
                              (a027748_row n) (a124010_row n)
-- Reinhard Zumkeller, Oct 03 2012

Maple

proc(n) local b,d: b := n^2: for d from 1 to n do if irem(n,d) = 0 and isprime(d) then b := b*(1+d^(-1)): fi: od: RETURN(b): end:

Mathematica

Table[ Fold[ If[ Mod[ n, #2 ]==0 && PrimeQ[ #2 ], #1*(1+1/#2), #1 ]&, n^2, Range[ n ] ], {n, 1, 45} ]
Table[ n^2 Times@@(1+1/Select[ Range[ 1, n ], (Mod[ n, #1 ]==0&&PrimeQ[ #1 ])& ]), {n, 1, 45} ] (* Olivier Gérard, Aug 15 1997 *)
f[p_, e_] := (p+1)*p^(2*e - 1); a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Jun 23 2020 *)

PARI

a(n)=if(n<1,0,direuler(p=2,n,(1+p*X)/(1-p^2*X))[n])

Formula

a(n) = n * A001615(n).

Dirichlet g.f.: zeta(s-1)*zeta(s-2)/zeta(2*s-2).

Dirichlet convolution: Sum_{d|n} mu(n/d)*sigma(d^2). - Vladeta Jovovic, Nov 16 2001

Multiplicative with a(p^e) = p^(2*e-1)*(p+1). - David W. Wilson, Aug 01 2001

a(n) = A181797(n)*A003557(n). - R. J. Mathar, Mar 30 2011

a(n) = A001615(n^2). - Enrique Pérez Herrero, Mar 06 2012

Sum_{k=1..n} a(k) ~ 5*n^3 / Pi^2. - Vaclav Kotesovec, Jan 11 2019

Sum_{n>=1} 1/a(n) = A335762. - Amiram Eldar, Jun 23 2020

Extensions

Additional comments from Michael Somos, May 19 2000