A319800 Numbers k such that Sum_{d|k} nphi(d) = k where the sum is over nonunitary divisors of k and nphi(k) is the nonunitary totient function (A254503).

3960, 5220, 1873080, 6733440, 8447040, 18685336320, 255306083760, 341863562880, 357274165248, 765899971200, 1018887932160

Offset

1,1

Comments

Ligh and Wall found the first 5 terms and also the terms 18685336320, 341863562880, 357174165248, 1018887932160, 20993596382889043200. They showed that each term has a powerful part with at least 2 distinct prime factors, and conjectured that it is only even.

References

Jozsef Sandor and Borislav Crstici, Handbook of Number Theory II, Kluwer Academic Publishers, 2004, Chapter 3, p. 287.

Links

Table of n, a(n) for n=1..11.

Steve Ligh and Charles R. Wall, Functions of Nonunitary Divisors, Fibonacci Quarterly, Vol. 25 (1987), pp. 333-338.

Mathematica

rad[n_] := Times @@ First /@ FactorInteger[n]; powerFree[n_] := Denominator[ n/rad[n]^2 ]; powerPart[n_] := n/powerFree[n]; nuphi[n_] := powerFree[ n ] * EulerPhi[powerPart[n]]; ndiv[n_] := Block[{d = Divisors[n]}, Select[d, GCD[#, n/#] > 1 &]]; a[n_] := Module[{d = ndiv[n]}, Total@Map[nuphi, d]]; s={}; Do[ If[a[n] == n, AppendTo[s, n]], {n, 1, 10^8}]; s

Prog

(PARI) nphi(n) = sumdiv(n, d, if(gcd(n/d, d) == 1, moebius(d)^2 * eulerphi(n/d)));
isok(n) = sumdiv(n, d, if(gcd(n/d, d) != 1, nphi(d))) == n; \\ Michel Marcus, Sep 28 2018

Crossrefs

Cf. A057521, A055231, A254503.

Sequence in context: A230707 A230617 A250531 * A373391 A034599 A007038

Adjacent sequences: A319797 A319798 A319799 * A319801 A319802 A319803

Keyword

nonn,more

Author

Amiram Eldar, Sep 28 2018

Extensions

a(6)-a(9) from Giovanni Resta, Sep 29 2018

a(10)-a(11) from Giovanni Resta, Oct 11 2018

Status

approved