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, 16733804567040, 19602402019200, 21205959667200, 79205761958400, 166967788700160, 189585719769600, 279604120561920, 623380501094400

Offset: 1

Author

Amiram Eldar, Sep 28 2018

Keywords

• nonn

Comments

Ligh and Wall found the first 6 terms and also the terms a(8) = 341863562880, a(9) = 357174165248, a(11) = 1018887932160, and 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

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

Crossrefs

Cf. A057521, A055231, A254503.

Programs

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

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

Extensions

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

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

a(12)-a(19) from Max Alekseyev, Jun 05 2025