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A275258 Toth's partial sum over the number of divisors of the greatest unitary divisor.

%I A275258 #18 Dec 22 2023 08:05:18
%S A275258 1,3,4,6,6,11,8,11,11,16,12,21,14,21,23,20,18,29,20,32,30,31,24,39,27,
%T A275258 36,30,42,30,57,32,37,45,46,47,56,38,51,52,59,42,77,44,62,63,61,48,71,
%U A275258 51,69,67,72,54,77,70,78,74,76,60,113,62,81,83
%N A275258 Toth's partial sum over the number of divisors of the greatest unitary divisor.
%H A275258 Amiram Eldar, <a href="/A275258/b275258.txt">Table of n, a(n) for n = 1..10000</a>
%H A275258 László Tóth, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL12/Toth2/toth5.html">On the Bi-Unitary Analogues of Euler's Arithmetical Function and the Gcd-Sum Function</a>, JIS 12 (2009), Article 09.5.2, function S**(n).
%F A275258 a(n) = Sum_{k=1..n} A000005( A165430(n,k) ).
%F A275258 Sum_{k=1..n} a(k) = c * n^2 / 2 + O(n * log(n)^2), where c = A065486. - _Amiram Eldar_, Dec 22 2023
%p A275258 A275258 := proc(n)
%p A275258     local a,d ;
%p A275258     a := 0 ;
%p A275258     for  d in A077610(n) do
%p A275258         a := a+A005361(d)*A275257(n/d,d) ;
%p A275258     end do:
%p A275258     a ;
%p A275258 end proc:
%p A275258 seq(A275258(n),n=1..80) ;
%t A275258 beta[n_] := Times @@ Transpose[FactorInteger[n]][[2]]; phi[x_, n_] := Sum[Boole[ GCD[k, n] == 1 ], {k, 1, x}]; a[n_] := DivisorSum[n, beta[#] * phi[n/#, #] &, GCD[#, n/#] == 1 &]; Array[a, 100] (* _Amiram Eldar_, Sep 22 2019 *)
%Y A275258 Cf. A000005, A065486, A165430.
%K A275258 nonn,look,easy
%O A275258 1,2
%A A275258 _R. J. Mathar_, Jul 21 2016