A064971 - OEIS

%I #19 Sep 20 2020 08:18:14

%S 1,6,12,20,30,72,56,72,90,180,132,240,182,336,360,272,306,540,380,600,

%T 672,792,552,864,650,1092,756,1120,870,2160,992,1056,1584,1836,1680,

%U 1800,1406,2280,2184,2160,1722,4032,1892,2640,2700,3312,2256

%N a(n) = n*usigma(n), where usigma(n) is the sum of unitary divisors of n (A034448).

%H Harry J. Smith, <a href="/A064971/b064971.txt">Table of n, a(n) for n = 1..1000</a>

%F Multiplicative with a(p^e) = p^e*(p^e+1). - _Vladeta Jovovic_, Nov 01 2001

%F Dirichlet g.f.: zeta(s-1)*zeta(s-2)/zeta(2*s-3). - _R. J. Mathar_, Feb 09 2011

%F Sum_{k=1..n} a(k) ~ Pi^2 * n^3 / (18*Zeta(3)). - _Vaclav Kotesovec_, Feb 01 2019

%F Sum_{k>=1} 1/a(k) = Product_{primes p} (3/2 + 1/(p-1) - (log(1-p) + QPolyGamma(1 - i*Pi/log(p), p))/log(p)) = 1.46909915920728851157169314962365889937120909118052326761431400799664418179... - _Vaclav Kotesovec_, Sep 20 2020

%p seq(mul(ifactors(n)[2][i][1]^ifactors(n)[2][i][2]*(1+ifactors(n)[2][i][1]^ifactors(n)[2][i][2]),i=1..nops(ifactors(n)[2])), n=1..50);

%o (PARI) usigma(n)= { local(f,s=1); f=factor(n); for(i=1, matsize(f)[1], s*=1 + f[i, 1]^f[i, 2]); return(s) } { for (n=1, 1000, write("b064971.txt", n, " ", n*usigma(n)) ) } \\ _Harry J. Smith_, Oct 01 2009

%Y Cf. A034448.

%K mult,nonn

%O 1,2

%A _N. J. A. Sloane_, Oct 30 2001