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a(n) begins the first run of least n consecutive numbers whose phi values have the same set of distinct prime divisors.
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%I #8 May 13 2018 04:36:06

%S 1,1,3,3,35,43570803,22154517001

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

%C 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).

%e a(5) = 35 since it is the least number such that

%e phi(35) = 24 = 2^3 * 3^1,

%e phi(36) = 12 = 2^2 * 3^1,

%e phi(37) = 36 = 2^2 * 3^2,

%e phi(38) = 18 = 2^1 * 3^2,

%e phi(39) = 24 = 2^3 * 3^1,

%e all having the same set of prime divisors: 2 and 3.

%t 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

%Y Cf. A000010, A001274, A007947, A303693.

%K nonn,more

%O 1,3

%A _Amiram Eldar_, May 01 2018

%E a(7) from _Giovanni Resta_, May 08 2018