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A291320
Numbers k such that uphi(k) is equal to the sum of the proper unitary divisors of k.
0
2, 600, 25584, 97464, 826560, 1249920, 50725248, 1372734720, 702637447680
OFFSET
1,1
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
EXAMPLE
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.
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 *)
PROG
(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
CROSSREFS
KEYWORD
nonn,more
AUTHOR
Altug Alkan, Aug 22 2017
EXTENSIONS
a(8) from Giovanni Resta, Aug 22 2017
a(9) from Giovanni Resta, May 12 2020
STATUS
approved