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A306070 Partial sums of A116550: Sum_{k=1..n} bphi(k) where bphi(k) is the bi-unitary analog of Euler's totient function.

%I A306070 #19 Jul 01 2025 01:03:06
%S A306070 1,2,4,7,11,14,20,27,35,41,51,59,71,80,89,104,120,132,150,164,178,193,
%T A306070 215,232,256,274,300,321,349,364,394,425,448,472,497,526,562,589,617,
%U A306070 648,688,709,751,786,820,853,899,935,983,1019,1056,1098,1150,1189
%N A306070 Partial sums of A116550: Sum_{k=1..n} bphi(k) where bphi(k) is the bi-unitary analog of Euler's totient function.
%C A306070 The bi-unitary version of A002088 and A177754.
%H A306070 Amiram Eldar, <a href="/A306070/b306070.txt">Table of n, a(n) for n = 1..10000</a>
%H A306070 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>, Journal of Integer Sequences, Vol. 12 (2009), Article 09.5.2.
%F A306070 a(n) = A*n^2/2 + O(n*log(n)^2), where A = A306071.
%t A306070 bphi[1] = 1; bphi[n_] := With[{pp = Power @@@ FactorInteger[n]}, Count[Range[n], m_ /; Intersection[pp, Power @@@ FactorInteger[m]] == {}]]; Accumulate[Table[bphi[n], {n, 1, 100}]] (* after _Jean-François Alcover_ at A116550 *)
%t A306070 phi[x_, n_] := DivisorSum[n, MoebiusMu[#]*Floor[x/#] &]; bphi[n_] := DivisorSum[n, (-1)^PrimeNu[#]*phi[n/#, #] &, CoprimeQ[#, n/#] &]; Accumulate[Array[bphi, 100]] (* _Amiram Eldar_, Jun 30 2025 *)
%o A306070 (PARI) udivs(n) = {my(d = divisors(n)); select(x->(gcd(x, n/x)==1), d); }
%o A306070 gcud(n, m) = vecmax(setintersect(udivs(n), udivs(m)));
%o A306070 bphi(n) = if (n==1, 1, sum(k=1, n-1, gcud(n, k) == 1));
%o A306070 a(n) = sum(k=1, n, bphi(k)); \\ _Michel Marcus_, Jun 20 2018
%Y A306070 Cf. A002088, A177754, A116550, A306071.
%K A306070 nonn
%O A306070 1,2
%A A306070 _Amiram Eldar_, Jun 19 2018