OFFSET
1,2
COMMENTS
Odd terms from A222264, see there for motivation and further links. In particular, the powers of 2 in A222264 which correspond to perfect numbers are excluded, so all n*x from this sequence are spoof perfect numbers, cf. A174292.
Here we do not exclude n if gcd(n,x) > 1 (the first such term is 1155), which would arguably be a "reasonable" additional condition to impose.
The first term with odd x is n=a(79)=9018009, x=22021, which yields Descarte's n*x=198585576189, see also A033870, A033871 and A222262.
No other term with odd x (and thus no other odd freestyle perfect number) is known as of today, to our best knowledge. See the paper by Banks et al. for some restrictions on such numbers.
One can note that when x is even, then sigma(n)/n is of the form (2k-2)/k. For instance, for n=15 we have x=4, and sigma(n)/n = 8/5 with k=5. On the other hand, when x is odd, then sigma(n)/n is of the form (2k-1)/k. For instance, for n=9018009 we have x=22021, and sigma(n)/n = 22021/11011 with k=11011. - Michel Marcus, Nov 24 2013
LINKS
Laszlo L. Toth, Table of n, a(n) for n = 1..692 (terms 1..143 terms from M. F. Hasler, 144..500 from Donovan Johnson).
W. Banks, A. Guloglu, W. Nevans and F. Saidak, Descartes Numbers, in: Anatomy of Integers, 167-174, American Mathematical Society, Providence R.I., 2008.
László Tóth, On the Density of Spoof Odd Perfect Numbers, arXiv:2101.09718 [math.NT], 2021.
EXAMPLE
For a(3)=15, we have sigma(15)=sigma(3*5)=4*6=24, thus 2*15/sigma(15) = 30/24 = 1+1/4. Here, x=4, and if 4 was assumed to be prime, then sigma'(4)=1+4=5 (*) and sigma'(15*4) = sigma(15)*sigma'(4) = 24*5 = 120 = 2*(15*4), i.e. 15*4 is a spoof perfect number.
(*) Here, sigma' is the "freestyle" sum-of-divisors function which assumes that 4 is prime, i.e., sigma'(n)=sigma(n/4^k)*(4^(k+1)-1)/3 with 4^k || n, or k=valuation(n,4)=floor(A007814(n)/2).
PROG
(PARI) forstep(n=1, 9e9, 2, numerator(2*n/sigma(n)-1)==1 & print1(n", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
M. F. Hasler, Feb 20 2013
STATUS
approved