A292818 Numbers n such that psi(k) - phi(k) = 2*n has no solution.

6, 51, 57, 65, 77, 87, 93, 95, 117, 119, 123, 145, 147, 155, 161, 171, 177, 185, 187, 189, 203, 205, 207, 209, 215, 217, 219, 221, 237, 245, 247, 249, 255, 261, 267, 275, 287, 291, 297, 299, 301, 303, 305, 321, 325, 327, 329, 335, 341, 345, 357, 363, 365, 371, 377, 387

Offset: 1

Author

Altug Alkan, Sep 24 2017

Keywords

• nonn

Comments

Inspired by a comment from Robert G. Wilson v.

All terms are composite.

Initial examples of forms of psi(k) - phi(k) where p, q, r, t are primes and a, b, c, d >= 1 as below:

If k = p^a, then psi(k) - phi(k) = 2*k/p.

If k = p^a*q^b, then psi(k) - phi(k) = 2*k*(p + q)/(p*q).

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).

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).

Examples

			6 is a term because psi(k) - phi(k) = 12 has no solution for any possible form of k.
		

Crossrefs

Cf. A000010, A001615, A292786.

Programs

Mathematica

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 *)