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

15, 207, 1023, 2975, 19359, 147455, 1207359, 5017599, 2170814463, 58946212863, 1166333691001023, 36472996363223648799

Offset: 1

Author

Jason Earls, Aug 30 2001

Keywords

• nonn
• more

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)

In other words, numbers m such that sigma(m)/(m+1) = 3/2. - Michel Marcus, Jan 03 2023

Some other terms: 374444425895728906239999, 2315893253834522244855807, 946345423297942718248771143999, 3181974057764759411641233725579002844163668627480799. - Max Alekseyev, Jul 30 2025

Examples

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

Links

• Antal Bege and Kinga Fogarasi, Generalized perfect numbers, Acta Univ. Sapientiae, Math., 1 (2009), 73-82.

Crossrefs

Cf. A000203, A009194, A014567, A058959.

Programs

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

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 *)

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, ", ")))

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

a(11)-a(12) from Max Alekseyev, Jul 30 2025