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
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.
Programs
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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
Comments