cp's OEIS Frontend

A306775 Partial sums of A060648: sum of the inverse Moebius transform of the Dedekind psi function from 1 to n.

%I A306775 #14 Oct 10 2019 03:37:03
%S A306775 1,5,10,20,27,47,56,78,95,123,136,186,201,237,272,318,337,405,426,496,
%T A306775 541,593,618,728,765,825,878,968,999,1139,1172,1266,1331,1407,1470,
%U A306775 1640,1679,1763,1838,1992,2035,2215,2260,2390,2509,2609,2658,2888,2953,3101
%N A306775 Partial sums of A060648: sum of the inverse Moebius transform of the Dedekind psi function from 1 to n.
%C A306775 In general, for m >= 1, Sum_{k=1..n} Sum_{d|k} psi_m(d) = Sum_{k=1..n} k^m * A064608(floor(n/k)), where psi_m(d) is the generalized Dedekind psi function.
%C A306775 Additionally, for m >= 1, Sum_{k=1..n} Sum_{d|k} psi_m(d) ~ (n^(m+1) * zeta(m+1)^2) / ((m+1) * zeta(2*(m+1))).
%H A306775 Vaclav Kotesovec, <a href="/A306775/b306775.txt">Table of n, a(n) for n = 1..10000</a>
%H A306775 Wikipedia, <a href="https://en.wikipedia.org/wiki/Dedekind_psi_function">Dedekind psi function</a>
%F A306775 a(n) ~ (5/4) * n^2.
%F A306775 a(n) = Sum_{k=1..n} A060648(k).
%F A306775 a(n) = Sum_{k=1..n} Sum_{d|k} A001615(d).
%F A306775 a(n) = Sum_{k=1..n} k * A064608(floor(n/k)).
%F A306775 a(n) = (1/2)*Sum_{k=1..n} 2^omega(k) * floor(n/k) * floor(1+n/k).
%F A306775 a(n) = Sum_{k=1..n} A001615(k)*floor(n/k). - _Ridouane Oudra_, Aug 27 2019
%p A306775 with(numtheory): psi := n -> n*mul(1+1/p, p in factorset(n)):
%p A306775 seq(add(psi(i)*floor(n/i), i=1..n), n=1..80); # _Ridouane Oudra_, Aug 27 2019
%t A306775 Accumulate[Table[Sum[EulerPhi[n/d] * DivisorSigma[0, d^2], {d, Divisors[n]}], {n, 1, 100}]] (* _Vaclav Kotesovec_, Oct 09 2019 *)
%o A306775 (PARI) a(n) = sum(k=1, n, 2^omega(k) * (n\k) * (1+n\k))/2;
%Y A306775 Cf. A001221, A001615, A034444, A060648, A061503, A064608.
%K A306775 nonn
%O A306775 1,2
%A A306775 _Daniel Suteu_, Mar 09 2019