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%I #46 Jun 23 2020 06:17:00
%S 1,6,12,24,30,72,56,96,108,180,132,288,182,336,360,384,306,648,380,
%T 720,672,792,552,1152,750,1092,972,1344,870,2160,992,1536,1584,1836,
%U 1680,2592,1406,2280,2184,2880,1722,4032,1892,3168,3240,3312,2256
%N a(n) = n^2*Product_{p|n} (1 + 1/p).
%C For n > 1: A006530(a(n)) = A076566(n-1). - _Reinhard Zumkeller_, Oct 03 2012
%C 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
%D B. Schoeneberg, Elliptic Modular Functions, Springer-Verlag, NY, 1974, p. 79.
%H T. D. Noe, <a href="/A000082/b000082.txt">Table of n, a(n) for n = 1..1000</a>
%H <a href="/index/Di#divseq">Index to divisibility sequences</a>
%F a(n) = n * A001615(n).
%F Dirichlet g.f.: zeta(s-1)*zeta(s-2)/zeta(2*s-2).
%F Dirichlet convolution: Sum_{d|n} mu(n/d)*sigma(d^2). - _Vladeta Jovovic_, Nov 16 2001
%F Multiplicative with a(p^e) = p^(2*e-1)*(p+1). - _David W. Wilson_, Aug 01 2001
%F a(n) = A181797(n)*A003557(n). - _R. J. Mathar_, Mar 30 2011
%F a(n) = A001615(n^2). - _Enrique Pérez Herrero_, Mar 06 2012
%F Sum_{k=1..n} a(k) ~ 5*n^3 / Pi^2. - _Vaclav Kotesovec_, Jan 11 2019
%F Sum_{n>=1} 1/a(n) = A335762. - _Amiram Eldar_, Jun 23 2020
%p 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:
%t Table[ Fold[ If[ Mod[ n, #2 ]==0 && PrimeQ[ #2 ], #1*(1+1/#2), #1 ]&, n^2, Range[ n ] ], {n, 1, 45} ]
%t Table[ n^2 Times@@(1+1/Select[ Range[ 1, n ], (Mod[ n, #1 ]==0&&PrimeQ[ #1 ])& ]), {n, 1, 45} ] (* _Olivier Gérard_, Aug 15 1997 *)
%t 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 *)
%o (PARI) a(n)=if(n<1,0,direuler(p=2,n,(1+p*X)/(1-p^2*X))[n])
%o (Haskell)
%o a000082 n = product $ zipWith (\p e -> p ^ (2*e - 1) * (p + 1))
%o (a027748_row n) (a124010_row n)
%o -- _Reinhard Zumkeller_, Oct 03 2012
%Y Cf. A001615, A027748, A033196, A124010, A335762.
%K nonn,easy,nice,mult
%O 1,2
%A _N. J. A. Sloane_
%E Additional comments from _Michael Somos_, May 19 2000
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