A303820 a(n) begins the first run of least n consecutive numbers whose phi values have the same set of distinct prime divisors.

1, 1, 3, 3, 35, 43570803, 22154517001

Offset: 1

Author

Amiram Eldar, May 01 2018

Keywords

• nonn
• more

Comments

a(n) is the least k such that rad(phi(k)) = rad(phi(k+1)) = ... = rad(phi(k+n-1)), where rad(n) is the squarefree kernel of n (A007947) and phi(n) is the Euler totient function of n (A000010).

Examples

			a(5) = 35 since it is the least number such that
phi(35) = 24 = 2^3 * 3^1,
phi(36) = 12 = 2^2 * 3^1,
phi(37) = 36 = 2^2 * 3^2,
phi(38) = 18 = 2^1 * 3^2,
phi(39) = 24 = 2^3 * 3^1,
all having the same set of prime divisors: 2 and 3.
		

Crossrefs

Cf. A000010, A001274, A007947, A303693.

Programs

Mathematica

rad[n_] := Times @@ (First@# & /@ FactorInteger@n); radphi[n_] := rad[ EulerPhi[n] ]; Seq[n_, q_] := Map[radphi, Range[n, n + q - 1]]; findConsec[q_, nmin_, nmax_] := Module[{}, s = Seq[1, q]; n = q + 1; Do[If[CountDistinct[s] == 1, Break[]]; s = Rest[AppendTo[s, radphi[n]]]; n++, {k, nmin, nmax}]; n - q]; seq = {1}; nmax = 10^10; Do[n1 = Last[ seq ]; s1 = findConsec[m, n1, nmax]; AppendTo[seq, s1], {m, 2, 6}]; seq

Extensions

a(7) from Giovanni Resta, May 08 2018