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A063906 Numbers m such that m = 2*sigma(m)/3 - 1.

%I A063906 #63 Aug 01 2025 00:10:42
%S A063906 15,207,1023,2975,19359,147455,1207359,5017599,2170814463,58946212863,
%T A063906 1166333691001023,36472996363223648799
%N A063906 Numbers m such that m = 2*sigma(m)/3 - 1.
%C A063906 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 A063906 From _Robert Israel_, Jan 12 2016: (Start)
%C A063906 All terms are odd and satisfy A009194(m) = 1 or 3.
%C A063906 Includes 3^(k-1)*(3^k-4) for k in A058959. The first few terms of this form are 15, 207, 19359, 36472996363223648799.
%C A063906 Other terms include 3^15*43048567*1003302465131 = 619739816695811335405066239 and 3^15*43049011*808868950607 = 499643410492503517919703039. (End)
%C A063906 In other words, numbers m such that sigma(m)/(m+1) = 3/2. - _Michel Marcus_, Jan 03 2023
%C A063906 Some other terms: 374444425895728906239999, 2315893253834522244855807, 946345423297942718248771143999, 3181974057764759411641233725579002844163668627480799. - _Max Alekseyev_, Jul 30 2025
%H A063906 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 A063906 sigma(1207359) = 1811040; 1811040 - 1207359 - 1 = 603680; abs(603680 - 1207359) + 1 = 603680.
%p A063906 select(n -> numtheory:-sigma(n) = 3/2*(n+1), [seq(i,i=1..10^6,2)]); # _Robert Israel_, Jan 12 2016
%t A063906 Select[Range[10^6], 2 * DivisorSigma[1, #]/3 - 1 == # &] (* _Giovanni Resta_, Apr 14 2016 *)
%o A063906 (PARI) s(n) = sigma(n)-n-1;
%o A063906 t(n) = abs(s(n)-n)+1;
%o A063906 for(n=1,10^8, if(t(n)==s(n),print1(n, ", ")))
%o A063906 (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 A063906 (Magma) [n: n in [1..6*10^6] | 2*DivisorSigma(1,n)/3-1 eq n]; // _Vincenzo Librandi_, Oct 10 2017
%Y A063906 Cf. A000203, A009194, A014567, A058959.
%K A063906 nonn,more
%O A063906 1,1
%A A063906 _Jason Earls_, Aug 30 2001
%E A063906 More terms from _Klaus Brockhaus_, Sep 01 2001
%E A063906 a(9)-a(10) from _Giovanni Resta_, Apr 14 2016
%E A063906 Simpler title suggested by _Giovanni Resta_, Apr 14 2016, based on formula provided by _Paolo P. Lava_, Jan 12 2016
%E A063906 a(11)-a(12) from _Max Alekseyev_, Jul 30 2025