A063906 - OEIS

%I #52 Jan 03 2023 09:23:51

%S 15,207,1023,2975,19359,147455,1207359,5017599,2170814463,58946212863

%N Numbers m such that m = 2*sigma(m)/3 - 1.

%C Original title: numbers n such that t(n) = s(n), where s(n) = sigma(n)-n-1 and t(n) = |s(n)-n|+1.

%C From _Robert Israel_, Jan 12 2016: (Start)

%C All terms are odd and satisfy A009194(m) = 1 or 3.

%C Includes 3^(k-1)*(3^k-4) for k in A058959.

%C The first few terms of this form are 15, 207, 19359, 36472996363223648799.

%C Other terms include 3^15*43048567*1003302465131 = 619739816695811335405066239 and 3^15*43049011*808868950607 = 499643410492503517919703039. (End)

%C a(11) > 10^12. - _Giovanni Resta_, Apr 14 2016

%C In other words, numbers m such that sigma(m)/(m+1) = 3/2. - _Michel Marcus_, Jan 03 2023

%H Antal Bege and Kinga Fogarasi, <a href="http://www.acta.sapientia.ro/acta-math/C1-1/MATH1-6.PDF">Generalized perfect numbers</a>, Acta Univ. Sapientiae, Math., 1 (2009), 73-82.

%e sigma(1207359) = 1811040; 1811040 - 1207359 - 1 = 603680; abs(603680 - 1207359) + 1 = 603680.

%p select(n -> numtheory:-sigma(n) = 3/2*(n+1), [seq(i,i=1..10^6,2)]); # _Robert Israel_, Jan 12 2016

%t Select[Range[10^6], 2 * DivisorSigma[1, #]/3 - 1 == # &] (* _Giovanni Resta_, Apr 14 2016 *)

%o (PARI) s(n) = sigma(n)-n-1;

%o t(n) = abs(s(n)-n)+1;

%o for(n=1,10^8, if(t(n)==s(n),print1(n, ", ")))

%o (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;

%o (Magma) [n: n in [1..6*10^6] | 2*DivisorSigma(1,n)/3-1 eq n]; // _Vincenzo Librandi_, Oct 10 2017

%Y Cf. A000203, A009194, A014567, A058959.

%K nonn,more

%O 1,1

%A _Jason Earls_, Aug 30 2001

%E More terms from _Klaus Brockhaus_, Sep 01 2001

%E a(9)-a(10) from _Giovanni Resta_, Apr 14 2016

%E Simpler title suggested by _Giovanni Resta_, Apr 14 2016, based on formula provided by _Paolo P. Lava_, Jan 12 2016