A298759 Numbers k such that bphi(k) = k/2, where bphi is the bi-unitary analog of Euler's totient function (A116550).

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

Offset: 1

Author

Amiram Eldar, Jan 26 2018

Keywords

• nonn
• more

Comments

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.

a(8) > 2*10^9, if it exists. - Amiram Eldar, Jul 16 2022

Examples

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

Crossrefs

Cf. A047994, A030163, A116550.

Programs

Mathematica

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]

PARI

udivs(n) = {my(d = divisors(n)); select(x->(gcd(x, n/x)==1), d); }
gcud(n, m) = vecmax(setintersect(udivs(n), udivs(m)));
bphi(n) = if (n==1, 1, sum(k=1, n-1, gcud(n, k) == 1));
isok(n) = bphi(n) == n/2; \\ Michel Marcus, Jan 26 2018

Extensions

a(7) from Amiram Eldar, Jul 16 2022