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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).

%I A319800 #31 Jun 05 2025 12:33:48
%S A319800 3960,5220,1873080,6733440,8447040,18685336320,255306083760,
%T A319800 341863562880,357274165248,765899971200,1018887932160,16733804567040,
%U A319800 19602402019200,21205959667200,79205761958400,166967788700160,189585719769600,279604120561920,623380501094400
%N 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).
%C A319800 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.
%D A319800 Jozsef Sandor and Borislav Crstici, Handbook of Number Theory II, Kluwer Academic Publishers, 2004, Chapter 3, p. 287.
%H A319800 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 A319800 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 A319800 (PARI) nphi(n) = sumdiv(n, d, if(gcd(n/d, d) == 1, moebius(d)^2 * eulerphi(n/d)));
%o A319800 isok(n) = sumdiv(n, d, if(gcd(n/d, d) != 1, nphi(d))) == n; \\ _Michel Marcus_, Sep 28 2018
%Y A319800 Cf. A057521, A055231, A254503.
%K A319800 nonn
%O A319800 1,1
%A A319800 _Amiram Eldar_, Sep 28 2018
%E A319800 a(6)-a(9) from _Giovanni Resta_, Sep 29 2018
%E A319800 a(10)-a(11) from _Giovanni Resta_, Oct 11 2018
%E A319800 a(12)-a(19) from _Max Alekseyev_, Jun 05 2025