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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).
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%I #27 Jan 05 2025 19:51:41

%S 3960,5220,1873080,6733440,8447040,18685336320,255306083760,

%T 341863562880,357274165248,765899971200,1018887932160

%N 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).

%C 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.

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

%H Steve Ligh and Charles R. Wall, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/25-4/ligh.pdf">Functions of Nonunitary Divisors</a>, Fibonacci Quarterly, Vol. 25 (1987), pp. 333-338.

%t 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

%o (PARI) nphi(n) = sumdiv(n, d, if(gcd(n/d, d) == 1, moebius(d)^2 * eulerphi(n/d)));

%o isok(n) = sumdiv(n, d, if(gcd(n/d, d) != 1, nphi(d))) == n; \\ _Michel Marcus_, Sep 28 2018

%Y Cf. A057521, A055231, A254503.

%K nonn,more

%O 1,1

%A _Amiram Eldar_, Sep 28 2018

%E a(6)-a(9) from _Giovanni Resta_, Sep 29 2018

%E a(10)-a(11) from _Giovanni Resta_, Oct 11 2018