A293184 Numbers k such that bphi(k) = bphi(k+1), where bphi(k) is the bi-unitary analog of Euler's totient function (A116550).

1, 14, 20, 57, 187, 188, 916, 1603, 93928, 142891, 432976, 549815, 692259, 773887, 872191, 4297168, 9478088, 127162432, 127991488, 129015616, 132527167

Offset: 1

Author

Amiram Eldar, Oct 01 2017

Keywords

• nonn
• more

Comments

187 is the first solution to bphi(k) = bphi(k+1) = bphi(k+2).

a(22) > 1.6*10^9, if it exists. - Amiram Eldar, Jul 16 2022

Examples

			14 is in the sequence since bphi(14) = bphi(15) = 9.
		

Crossrefs

Cf. A001274, A116550, A287055, A294030.

Programs

Mathematica

bphi[1] = 1; bphi[n_] := With[{pp = Power @@@ FactorInteger[n]},   Count[Range[n], m_ /; Intersection[pp, Power @@@ FactorInteger[m]] == {}]]; a={}; b1=0; Do[b2 = bphi[k];If[b1 == b2, a = AppendTo[a, k - 1]]; b1 = b2, {k, 1, 10^3}]; a (* after Jean-François Alcover at A116550 *)

PARI

udivs(n) = {my(d = divisors(n)); select(x->(gcd(x, n/x)==1), d); }
gcud(n, m) = vecmax(setintersect(udivs(n), udivs(m)));
biuphi(n) = if (n==1, 1, sum(k=1, n-1, gcud(n, k) == 1));
isok(n) = biuphi(n) == biuphi(n+1);
lista(nn) = {x = biuphi(1); for (n=2, nn, y = biuphi(n); if (x==y, print1(n-1, ", ")); x = y;);} \\ Michel Marcus, Nov 09 2017

Extensions

a(10) from Michel Marcus, Nov 11 2017

a(11) from Michel Marcus, Nov 12 2017

a(12)-a(21) from Amiram Eldar, Jul 16 2022