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%I #13 Oct 04 2017 14:13:17
%S 6,51,57,65,77,87,93,95,117,119,123,145,147,155,161,171,177,185,187,
%T 189,203,205,207,209,215,217,219,221,237,245,247,249,255,261,267,275,
%U 287,291,297,299,301,303,305,321,325,327,329,335,341,345,357,363,365,371,377,387
%N Numbers n such that psi(k) - phi(k) = 2*n has no solution.
%C Inspired by a comment from _Robert G. Wilson v_.
%C All terms are composite.
%C Initial examples of forms of psi(k) - phi(k) where p, q, r, t are primes and a, b, c, d >= 1 as below:
%C If k = p^a, then psi(k) - phi(k) = 2*k/p.
%C If k = p^a*q^b, then psi(k) - phi(k) = 2*k*(p + q)/(p*q).
%C If k = p^a*q^b*r^c, then psi(k) - phi(k) = 2*k*(p*q + q*r + p*r + 1)/(p*q*r).
%C If k = p^a*q^b*r^c*t^d, then psi(k) - phi(k) = 2*k*(p*q*r + p*q*t + p*r*t + q*r*t + p + q + r + t)/(p*q*r*t).
%e 6 is a term because psi(k) - phi(k) = 12 has no solution for any possible form of k.
%t psi[n_] := If[n == 1, 1, n Times @@ (1 + 1/First /@ FactorInteger@ n)]; upto[n_] := Block[{d, T = 0 Range[n]}, Do[d = (psi[k] - EulerPhi[k])/2; If[d <= n, T[[d]] = 1], {k, 2, n^2}]; Flatten@ Position[T, 0]]; upto[387] (* _Giovanni Resta_, Sep 25 2017 *)
%Y Cf. A000010, A001615, A292786.
%K nonn
%O 1,1
%A _Altug Alkan_, Sep 24 2017
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