A298759 - OEIS

%I #11 Jul 16 2022 07:14:03

%S 2,6,30,42,1722,1806,19977474

%N Numbers k such that bphi(k) = k/2, where bphi is the bi-unitary analog of Euler's totient function (A116550).

%C With Euler's totient function, phi(k) = k/2 only for powers of 2 (A000079, except for 1). With the unitary totient function (A047994) the corresponding sequence is A030163.

%C a(8) > 2*10^9, if it exists. - _Amiram Eldar_, Jul 16 2022

%e 42 is in the sequence since bphi(42) = 21 = 42/2.

%t bphi[1] = 1; bphi[n_] :=  With[{pp = Power @@@ FactorInteger[n]},   Count[Range[n], m_ /; Intersection[pp, Power @@@ FactorInteger[m]] == {}]]; aQ[n_] := bphi[n] == n/2; Select[Range[10000], aQ]

%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 isok(n) = bphi(n) == n/2; \\ _Michel Marcus_, Jan 26 2018

%Y Cf. A047994, A030163, A116550.

%K nonn,more

%O 1,1

%A _Amiram Eldar_, Jan 26 2018

%E a(7) from _Amiram Eldar_, Jul 16 2022