OFFSET
1,1
COMMENTS
Original title: numbers n such that t(n) = s(n), where s(n) = sigma(n)-n-1 and t(n) = |s(n)-n|+1.
From Robert Israel, Jan 12 2016: (Start)
All terms are odd and satisfy A009194(m) = 1 or 3.
Includes 3^(k-1)*(3^k-4) for k in A058959.
The first few terms of this form are 15, 207, 19359, 36472996363223648799.
Other terms include 3^15*43048567*1003302465131 = 619739816695811335405066239 and 3^15*43049011*808868950607 = 499643410492503517919703039. (End)
a(11) > 10^12. - Giovanni Resta, Apr 14 2016
In other words, numbers m such that sigma(m)/(m+1) = 3/2. - Michel Marcus, Jan 03 2023
LINKS
Antal Bege and Kinga Fogarasi, Generalized perfect numbers, Acta Univ. Sapientiae, Math., 1 (2009), 73-82.
EXAMPLE
sigma(1207359) = 1811040; 1811040 - 1207359 - 1 = 603680; abs(603680 - 1207359) + 1 = 603680.
MAPLE
select(n -> numtheory:-sigma(n) = 3/2*(n+1), [seq(i, i=1..10^6, 2)]); # Robert Israel, Jan 12 2016
MATHEMATICA
Select[Range[10^6], 2 * DivisorSigma[1, #]/3 - 1 == # &] (* Giovanni Resta, Apr 14 2016 *)
PROG
(PARI) s(n) = sigma(n)-n-1;
t(n) = abs(s(n)-n)+1;
for(n=1, 10^8, if(t(n)==s(n), print1(n, ", ")))
(ARIBAS): for n := 1 to 4000000 do s := sigma(n) - n - 1; t := abs(s - n) + 1; if s = t then write(n, " "); end; end;
(Magma) [n: n in [1..6*10^6] | 2*DivisorSigma(1, n)/3-1 eq n]; // Vincenzo Librandi, Oct 10 2017
CROSSREFS
KEYWORD
nonn,more
AUTHOR
Jason Earls, Aug 30 2001
EXTENSIONS
More terms from Klaus Brockhaus, Sep 01 2001
a(9)-a(10) from Giovanni Resta, Apr 14 2016
Simpler title suggested by Giovanni Resta, Apr 14 2016, based on formula provided by Paolo P. Lava, Jan 12 2016
STATUS
approved