A291320 Numbers k such that uphi(k) is equal to the sum of the proper unitary divisors of k.

2, 600, 25584, 97464, 826560, 1249920, 50725248, 1372734720, 702637447680

Offset: 1

Author

Altug Alkan, Aug 22 2017

Keywords

• nonn
• more

Comments

Or numbers k such that usigma(k) - k = uphi(k) where usigma(k) = A034448(k) and uphi(k) = A047994(k).

a(10) > 10^13. - Giovanni Resta, May 12 2020

Examples

			600 = 2^3*3*5^2 is a term because usigma(600) - uphi(600) = (2^3+1)*(3+1)*(5^2+1) - (2^3-1)*(3-1)*(5^2-1) = 600.
		

Crossrefs

Cf. A034448, A047994, A063919.

Programs

Mathematica

ok[n_] := Block[{p = Power @@@ FactorInteger[n]}, Times @@ (p + 1) == n + Times @@ (p - 1)]; Select[Range[2, 10^6], ok] (* Giovanni Resta, Aug 22 2017 *)

PARI

usigma(n) = sumdivmult(n, d, if(gcd(d, n/d)==1, d));
uphi(n) = my(f=factor(n)~); prod(i=1, #f, f[1, i]^f[2, i]-1);
isok(n) = usigma(n)-uphi(n)==n;

PARI

list(lim)=my(v=List()); forfactored(n=2,lim\1, if(sumdivmult(n, d, if(gcd(d, n[1]/d)==1, d))-prod(i=1, #n[2]~, n[2][i,1]^n[2][i,2]-1)==n[1], listput(v,n[1]))); Vec(v) \\ Charles R Greathouse IV, Aug 22 2017

Extensions

a(8) from Giovanni Resta, Aug 22 2017

a(9) from Giovanni Resta, May 12 2020