A200758 Superimperfect numbers.

2, 4, 8, 128, 32768, 2147483648

Offset: 1

Author

Laszlo Toth, Nov 22 2011

Keywords

• nonn
• more

Comments

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.

Links

• Laszlo Toth, A survey of the alternating sum-of-divisors function, arXiv:1111.4842 [math.NT], 2011-2014.
• Laszlo Toth, The alternating sum-of-divisors function, 9th Joint Conf. on Math. and Comp. Sci., February 9-12, 2012, Siofok, Hungary.

Crossrefs

Cf. A061020, A127724, A127725, A019279, A058891, A206369.

Programs

PARI

beta(n)=sumdiv(n,d,(-1)^bigomega(n/d)*d)
for(n=1,1e8,if(2*beta(beta(n))==n,print1(n", "))) \\ Charles R Greathouse IV, Nov 22 2011

PARI

ak(p,e)=my(s=1); for(i=1,e, s=s*p + (-1)^i); s
beta(n)=my(f=factor(n)); prod(i=1,#f~, ak(f[i,1],f[i,2]))
is(n)=my(b=beta(n)); 2*b-2 >= n && 2*beta(b)==n \\ Charles R Greathouse IV, Dec 27 2016