A291320 - OEIS

%I #23 Jul 24 2021 01:10:43

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

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

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

%C a(10) > 10^13. - _Giovanni Resta_, May 12 2020

%e 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.

%t 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 *)

%o (PARI) usigma(n) = sumdivmult(n, d, if(gcd(d, n/d)==1, d));

%o uphi(n) = my(f=factor(n)~); prod(i=1, #f, f[1, i]^f[2, i]-1);

%o isok(n) = usigma(n)-uphi(n)==n;

%o (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

%Y Cf. A034448, A047994, A063919.

%K nonn,more

%O 1,1

%A _Altug Alkan_, Aug 22 2017

%E a(8) from _Giovanni Resta_, Aug 22 2017

%E a(9) from _Giovanni Resta_, May 12 2020