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%I #56 Mar 08 2019 16:15:10
%S 2,9,165,238,1640,4320,10250,10824,13500,23760,58500,66912,425880,
%T 520128,873180,931392,1899744,2129400,2253888,3276000,4580064,4668300,
%U 13722800,15459840,40360320,201801600,439021440,3809332800,15359485680,794436968640,1407035080704
%N Unitary superperfect numbers: numbers n such that usigma(usigma(n)) = 2*n, where usigma(n) is the sum of unitary divisors of n (A034448).
%C May be called (2,2)-unitary perfect numbers, analogous to (k,l)-perfect numbers.
%C Sitaramaiah and Subbarao found the first 22 terms. Also in the sequence is 12189313382400. - _Amiram Eldar_, Feb 27 2019
%H V. Sitaramaiah and M. V. Subbarao, <a href="http://www.math.ualberta.ca/~subbarao/documents/Sitaramaiah_Subbarao1998.pdf">On the equation sigma*(sigma*(n)) = 2n</a>, Utilitas Mathematica, Vol. 53 (1998), pp. 101-124.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/SuperUnitaryPerfectNumber.html">Super Unitary Perfect Number.</a>
%H Tomohiro Yamada, <a href="http://mathematica-pannonica.ttk.pte.hu/index_elemei/mp19-1/MP19-1(2008)pp37-47.pdf">Unitary super perfect numbers</a>, Mathematica Pannonica, Volume 19, No. 1, 2008, pp. 37-47; <a href="https://arxiv.org/abs/0802.4377">Preprint</a>, arXiv:0802.4377 [math.NT], 2008. Proves that 9 and 165 are the only odd terms of this sequence.
%H Tomohiro Yamada, <a href="http://ac.inf.elte.hu/Vol_048_2018/247_48.pdf">2 and 9 are the only biunitary superperfect numbers</a>, Annales Univ. Sci. Budapest., Sec. Comp., Volume 48 (2018). Mentions this sequence.
%t 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 *)
%o (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);}
%o for(n=1,10^7, if(usigma(usigma(n))==2*n, print1(n, ", ")))
%Y Cf. A034448, A019279.
%Y Cf. A064012 (usigma(usigma(n)) = 3n).
%K nonn,nice
%O 1,1
%A _Yasutoshi Kohmoto_
%E Corrected by _Jason Earls_, Aug 25 2001
%E More terms from _Jud McCranie_, Oct 28 2001
%E Offset corrected and a(28) from _Donovan Johnson_, Jul 23 2012
%E Name edited and a(29) from _Amiram Eldar_, Feb 27 2019
%E a(30)-a(31) from _Giovanni Resta_, Mar 08 2019
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