A306070 - OEIS

%I #12 Jul 28 2018 11:50:41

%S 1,2,4,7,11,14,20,27,35,41,51,59,71,80,89,104,120,132,150,164,178,193,

%T 215,232,256,274,300,321,349,364,394,425,448,472,497,526,562,589,617,

%U 648,688,709,751,786,820,853,899,935,983,1019,1056,1098,1150,1189

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

%C The bi-unitary version of A002088 and A177754.

%H 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 a(n) = A*n^2/2 + O(n*log(n)^2), where A = A306071.

%t a[1] = 1; a[n_] := With[{pp = Power @@@ FactorInteger[n]}, Count[Range[n], m_ /; Intersection[pp, Power @@@ FactorInteger[m]] == {}]]; Accumulate[Table[a[n], {n, 1, 100}]] (* after _Jean-François Alcover_ at A116550 *)

%o (PARI) udivs(n) = {my(d = divisors(n)); select(x->(gcd(x, n/x)==1), d); }

%o gcud(n, m) = vecmax(setintersect(udivs(n), udivs(m)));

%o bphi(n) = if (n==1, 1, sum(k=1, n-1, gcud(n, k) == 1));

%o a(n) = sum(k=1, n, bphi(k)); \\ _Michel Marcus_, Jun 20 2018

%Y Cf. A002088, A177754, A116550, A306071.

%K nonn

%O 1,2

%A _Amiram Eldar_, Jun 19 2018