Search: id:A000082
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%I A000082
%S A000082 1,6,12,24,30,72,56,96,108,180,132,288,182,336,360,384,306,648,380,
%T A000082 720,672,792,552,1152,750,1092,972,1344,870,2160,992,1536,1584,1836,
%U A000082 1680,2592,1406,2280,2184,2880,1722,4032,1892,3168,3240,3312,2256
%N A000082 n^2*Product_{p|n} (1 + 1/p).
%D A000082 B. Schoeneberg, Elliptic Modular Functions, Springer-Verlag, NY, 1974, 
               p. 79.
%H A000082 T. D. Noe, <a href="b000082.txt">Table of n, a(n) for n=1..1000</a>
%F A000082 Dirichlet g.f.: zeta(s-1)*zeta(s-2)/zeta(2*s-2).
%F A000082 Dirichlet convolution: Sum_{d|n} mu(n/d)*sigma(d^2). - Vladeta Jovovic 
               (vladeta(AT)eunet.rs), Nov 16 2001
%F A000082 Multiplicative with a(p^e) = p^(2*e-1)*(p+1);. - David W. Wilson (davidwwilson(AT)comcast.net), 
               Aug 01, 2001.
%p A000082 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 A000082 Table[ Fold[ If[ Mod[ n, #2 ]==0 && PrimeQ[ #2 ], #1*(1+1/#2), #1 ]&, 
               n^2, Range[ n ] ], {n, 1, 45} ]
%t A000082 Table[ n^2 Times@@(1+1/Select[ Range[ 1, n ], (Mod[ n, #1 ]==0&&PrimeQ[ 
               #1 ])& ]), {n, 1, 45} ]
%o A000082 (PARI) a(n)=if(n<1,0,direuler(p=2,n,(1+p*X)/(1-p^2*X))[n])
%Y A000082 a(n)=n*A001615(n). Cf. A033196.
%Y A000082 Sequence in context: A071611 A119500 A110967 this_sequence A106697 A140522 
               A065218
%Y A000082 Adjacent sequences: A000079 A000080 A000081 this_sequence A000083 A000084 
               A000085
%K A000082 nonn,easy,nice,mult
%O A000082 1,2
%A A000082 N. J. A. Sloane (njas(AT)research.att.com).
%E A000082 Mathematica Program Aug 15 1997 (Olivier Gerard). Additional comments 
               from Michael Somos, May 19, 2000.

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