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%I #37 Nov 07 2017 03:04:04
%S 2,4,8,128,32768,2147483648
%N Superimperfect numbers.
%C A number n is said to be superimperfect if 2*beta(beta(n)) = n, where beta is the multiplicative function defined by beta(p^e) = p^e - p^(e-1) + p^(e-2) - ... + (-1)^e for every prime power p^e. The function beta is called the alternating sum-of-divisors function. Here beta(n) is the absolute value of A061020(n). There are no other superimperfect numbers up to 10^7. The number 2^(2^k-1) is superimperfect if and only if k=1,2,3,4,5.
%H Laszlo Toth, <a href="http://arxiv.org/abs/1111.4842">A survey of the alternating sum-of-divisors function</a>, arXiv:1111.4842 [math.NT], 2011-2014.
%H Laszlo Toth, <a href="http://macs.elte.hu/downloads/abstracts/MaCS_abs_Toth.pdf">The alternating sum-of-divisors function</a>, 9th Joint Conf. on Math. and Comp. Sci., February 9-12, 2012, Siofok, Hungary.
%o (PARI) beta(n)=sumdiv(n,d,(-1)^bigomega(n/d)*d)
%o for(n=1,1e8,if(2*beta(beta(n))==n,print1(n", "))) \\ _Charles R Greathouse IV_, Nov 22 2011
%o (PARI) ak(p,e)=my(s=1); for(i=1,e, s=s*p + (-1)^i); s
%o beta(n)=my(f=factor(n)); prod(i=1,#f~, ak(f[i,1],f[i,2]))
%o is(n)=my(b=beta(n)); 2*b-2 >= n && 2*beta(b)==n \\ _Charles R Greathouse IV_, Dec 27 2016
%Y Cf. A061020, A127724, A127725, A019279, A058891, A206369.
%K nonn,more
%O 1,1
%A _Laszlo Toth_, Nov 22 2011
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