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A160678 Numbers n whose abundancy is equal to 13/2; sigma(n)/n = 13/2.

%I A160678 #21 Feb 21 2017 11:53:10
%S A160678 170974031122008628879954060917200710847692800,
%T A160678 1893010442758976546037991125738431754692198400,
%U A160678 54361481238923605327597493185154939181072384000
%N A160678 Numbers n whose abundancy is equal to 13/2; sigma(n)/n = 13/2.
%C A160678 This sequence includes many terms but it is conjectured to be finite.
%H A160678 Michel Marcus, <a href="/A160678/b160678.txt">Table of n, a(n) for n = 1..307</a>
%H A160678 G. P. Michon, <a href="http://www.numericana.com/answer/numbers.htm#multiperfect">Multiperfect and hemiperfect integers</a>
%H A160678 G. P. Michon, <a href="http://www.numericana.com/answer/numbers.htm#multiplicative">Multiplicative functions</a>: Abundancy = sigma(n)/n
%H A160678 G. P. Michon and M. Marcus, <a href="http://www.numericana.com/data/hpn13.htm">Hemiperfect numbers of abundancy 13/2</a>
%H A160678 Walter Nissen, <a href="http://upforthecount.com/math/abundance.html">Abundancy: Some Resources </a>
%e A160678 a(1) = 2^23 3^9 5^2 7^5 11^5 13^2 17 19^3 31 37 43 61^2 97 181 241.
%e A160678 As the "sum of divisors" function (sigma) is a multiplicative function, sigma(a(1)) is the product of the values of sigma at the above prime powers, respectively given as follows, in factorized form:
%e A160678 sigma(a(1)) = (3^2 5 7 13 17 241) (2^2 11^2 61) (31) (2^3 3 19 43) (2^2 3^2 7 19 37) (3 61) (2 3^2) (2^3 5 181) (2^5) (2 19) (2^2 11) (3 13 97) (2 7 13) (2 7^2) (2 11^2).
%e A160678 a(1) belongs to the sequence because the latter product boils down to 13/2 times the former.
%o A160678 (PARI) is(n)=sigma(n,-1)==13/2 \\ _Charles R Greathouse IV_, Feb 21 2017
%Y A160678 Cf. A000203 (sigma function, sum of divisors), A141643 (abundancy = 5/2), A055153 (abundancy = 7/2), A141645 (abundancy = 9/2), A159271 (abundancy = 11/2), A159907 (half-integral abundancy, "hemiperfect numbers"), A088912 (least numbers of given half-integer abundancy). A007691 (multiperfect numbers, abundancy is an integer), A000396 (perfect numbers, abundancy = 2), A005101 (abundant numbers, abundancy is greater than 2), A005100 (deficient numbers, abundancy is less than 2).
%K A160678 nonn
%O A160678 1,1
%A A160678 _Gerard P. Michon_, Jun 06 2009