A038843 Unitary superperfect numbers: numbers n such that usigma(usigma(n)) = 2*n, where usigma(n) is the sum of unitary divisors of n (A034448).

2, 9, 165, 238, 1640, 4320, 10250, 10824, 13500, 23760, 58500, 66912, 425880, 520128, 873180, 931392, 1899744, 2129400, 2253888, 3276000, 4580064, 4668300, 13722800, 15459840, 40360320, 201801600, 439021440, 3809332800, 15359485680, 794436968640, 1407035080704

Offset: 1

Author

Yasutoshi Kohmoto

Keywords

• nonn
• nice

Comments

May be called (2,2)-unitary perfect numbers, analogous to (k,l)-perfect numbers.

Sitaramaiah and Subbarao found the first 22 terms. Also in the sequence is 12189313382400. - Amiram Eldar, Feb 27 2019

Links

• V. Sitaramaiah and M. V. Subbarao, On the equation sigma*(sigma*(n)) = 2n, Utilitas Mathematica, Vol. 53 (1998), pp. 101-124.
• Eric Weisstein's World of Mathematics, Super Unitary Perfect Number.
• Tomohiro Yamada, Unitary super perfect numbers, Mathematica Pannonica, Volume 19, No. 1, 2008, pp. 37-47; Preprint, arXiv:0802.4377 [math.NT], 2008. Proves that 9 and 165 are the only odd terms of this sequence.
• Tomohiro Yamada, 2 and 9 are the only biunitary superperfect numbers, Annales Univ. Sci. Budapest., Sec. Comp., Volume 48 (2018). Mentions this sequence.

Crossrefs

Cf. A034448, A019279.

Cf. A064012 (usigma(usigma(n)) = 3n).

Programs

Mathematica

usigma[n_] := Times @@ (Apply[ Power, FactorInteger[n], {1}] + 1); n = 1; A038843 = {}; While[n < 10^7, If[ usigma[ usigma[n] ] == 2n, Print[n]; AppendTo[ A038843, n] ]; n++]; A038843 (* Jean-François Alcover, Dec 07 2011 *)

PARI

{usigma(n,s=1,fac,i)= fac=factor(n); for(i=1,matsize(fac)[1], s=s*(1+fac[i,1]^fac[i,2]) ); return(s);}
for(n=1,10^7, if(usigma(usigma(n))==2*n, print1(n, ", ")))

Extensions

Corrected by Jason Earls, Aug 25 2001

More terms from Jud McCranie, Oct 28 2001

Offset corrected and a(28) from Donovan Johnson, Jul 23 2012

Name edited and a(29) from Amiram Eldar, Feb 27 2019

a(30)-a(31) from Giovanni Resta, Mar 08 2019