|
%I #37 Sep 08 2022 08:45:28
%S 22,130,184,1012,2272,18904,33664,70564,85936,100804,391612,527872,
%T 1090912,17619844,2147713024,6800695312,34360655872,549759483904,
%U 1661355408388,28502765343364,82994670582016,99249696661504,120646991405056,431202442356004,952413274955776
%N Numbers n whose abundance sigma(n) - 2n = -8. Numbers n whose deficiency is 8.
%C a(19) > 10^12. - _Donovan Johnson_, Dec 08 2011
%C a(20) > 10^13. - _Giovanni Resta_, Mar 29 2013
%C a(30) > 10^18. - _Hiroaki Yamanouchi_, Aug 21 2018
%C a(20) <= 36028797958488064 ~ 3.6*10^16. Indeed, if k is in A057195 then 2^(k-1)*A168415(k) is in this sequence, and k=28 yields this upper bound for a(20) which is in any case a term of this sequence. - _M. F. Hasler_, Apr 27 2015
%C If n is in this sequence and p a prime not dividing n, then np is abundant if and only if p < sigma(n)/8 = n/4-1. For all n=a(k) except {22, 70564, 100804, 17619844}, there is such a p near this limit, such that n*p is a primitive weird number (A002975; in A258882 for the terms mentioned in the preceding comment). - _M. F. Hasler_, Jul 20 2016
%C Any term x of this sequence can be combined with any term y of A088833 to satisfy the property (sigma(x)+sigma(y))/(x+y) = 2, which is a necessary (but not sufficient) condition for two numbers to be amicable. - _Timothy L. Tiffin_, Sep 13 2016
%C Is there any odd number in this sequence? Is it possible to prove the contrary? - _M. F. Hasler_, Feb 22 2017
%H Hiroaki Yamanouchi, <a href="/A125247/b125247.txt">Table of n, a(n) for n = 1..29</a>
%e The abundance of 22 = (1+2+11+22)-44 = -8
%t Select[Range[10^6], DivisorSigma[1, #] - 2 # == -8 &] (* _Michael De Vlieger_, Jul 21 2016 *)
%o (PARI) for(n=1,1000000,if(((sigma(n)-2*n)==-8),print1(n,",")))
%o (Magma) [n: n in [1..2*10^7] | (DivisorSigma(1,n)-2*n) eq - 8]; // _Vincenzo Librandi_, Jul 22 2016
%Y Cf. A033880, A088833 (abundance 8).
%K easy,nonn
%O 1,1
%A _Jason G. Wurtzel_, Nov 25 2006
%E a(13)-a(15) from _Klaus Brockhaus_, Nov 29 2006
%E a(16)-a(17) from _Donovan Johnson_, Dec 23 2008
%E a(18) from _Donovan Johnson_, Dec 08 2011
%E a(19) from _Giovanni Resta_, Mar 29 2013
%E a(20)-a(25) from _Hiroaki Yamanouchi_, Aug 21 2018
| 