A254520 - OEIS

%I #17 Sep 20 2020 09:22:35

%S 1,4,9,12,25,36,49,48,72,100,121,108,169,196,225,192,289,288,361,300,

%T 441,484,529,432,600,676,648,588,841,900,961,768,1089,1156,1225,864,

%U 1369,1444,1521,1200,1681,1764,1849,1452,1800,2116,2209,1728,2352,2400

%N Möbius transform of A034676.

%C The Dirichlet convolution of a(n) and sigma(n) is sigma(n^2).

%H Álvar Ibeas, <a href="/A254520/b254520.txt">Table of n, a(n) for n = 1..10000</a>

%F a(n) = n^2 * Sum_{d^2 | n} (moebius(d) / d^2).

%F Multiplicative with a(p) = p^2; a(p^e) = p^(2e) - p^(2e-2), for e > 1.

%F Dirichlet g.f.: zeta(s-2) / zeta(2s-2).

%F Sum_{k=1..n} a(k) ~ 30 * n^3 / Pi^4. - _Vaclav Kotesovec_, Jan 11 2019

%F Sum_{k>=1} 1/a(k) = Product_{primes p} (1 + 1/p^2 + 1/(p^2 - 1)^2) = 1.681923034881403168503816690236967736500606659628336043348190538886262268... - _Vaclav Kotesovec_, Sep 20 2020

%o (PARI) a(n) = n^2*sumdiv(n, d, if (issquare(d), moebius(sqrtint(d))/d)); \\ _Michel Marcus_, Feb 10 2015

%Y Cf. A000290, A000203, A034676, A065764.

%K mult,nonn

%O 1,2

%A _Álvar Ibeas_, Jan 31 2015