A200758 Superimperfect numbers.

2, 4, 8, 128, 32768, 2147483648

Offset

1,1

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

Table of n, a(n) for n=1..6.

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.

Prog

(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

Crossrefs

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

Sequence in context: A130693 A286523 A060815 * A110746 A084010 A203630

Adjacent sequences: A200755 A200756 A200757 * A200759 A200760 A200761

Keyword

nonn,more

Author

Laszlo Toth, Nov 22 2011

Status

approved