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Least x = a(n) such that number of common prime factors (ignoring multiplicity) of sigma(x) = A000203(x) and phi(x) = A000010(x) equals n.
3

%I #21 Aug 28 2019 10:35:13

%S 3,14,209,3596,41624,2003639,24206049,2562857198,57721363052

%N Least x = a(n) such that number of common prime factors (ignoring multiplicity) of sigma(x) = A000203(x) and phi(x) = A000010(x) equals n.

%C a(10) <= 6804704928496. - _Donovan Johnson_, Jun 15 2013

%F a(n) = min{x: A081396(x) = n}.

%e x = 209: sigma(209) = 240 = 2^4*3*5, phi(209) = 180 = 2^2*3^2*5, common factor set = {2,3,5}, so a(3) = 209.

%t ffi[x_] := Flatten[FactorInteger[x]] lf[x_] := Length[FactorInteger[x]] ba[x_] := Table[Part[ffi[x], 2*w-1], {w, 1, lf[x]}] t=Table[0, {10}]; Do[s=Length[Intersection[ba[EulerPhi[n]], ba[DivisorSigma[1, n]]]]; If[s<11&&t[[s]]==0, t[[s]]=n], {n, 1, 1000000}]; t

%o (PARI) a(n)=my(k=prod(i=1,n,prime(i))); while(omega(gcd(sigma(k),eulerphi(k)))!=n, k++); k \\ _Charles R Greathouse IV_, Feb 14 2013

%Y Cf. A000203, A000010, A081396.

%K nonn,more

%O 1,1

%A _Labos Elemer_, Mar 28 2003

%E a(6)-a(8) from _Donovan Johnson_, May 24 2009

%E a(9) from _Donovan Johnson_, Jun 14 2013