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

1,1

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

Table of n, a(n) for n=1..19.

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: A230617 A250531 A395857 * A373391 A007038 A034599

Adjacent sequences: A319797 A319798 A319799 * A319801 A319802 A319803

Keyword

nonn

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

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

Status

approved