OFFSET
1,1
COMMENTS
If k is in A057195, then 2^(k-1)*A168415(k) is in this sequence. - M. F. Hasler, Apr 27 2015; updated by Max Alekseyev, Jan 26 2026
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
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
Is there any odd number in this sequence? Is it possible to prove the contrary? - M. F. Hasler, Feb 22 2017
From Alexander Violette, Jan 26 2026: (Start)
Any odd term has to be of the form m^2*p^(4k+1) where p is congruent to 1 mod 4.
22660587056580139737907204, 282743071153587037728145408, 54399917187200570519051754035740672, 66461400903391833556072542653906944, 164399910729457037686170602275463956083792004, 305760004048037867437501799970114169564867072098304, and 5698066062066428054450747296180128604174368806779427872765261055600361472 are terms. (End)
Also contains 2711439023398152263138022154548783234854501343574298189627392. - Max Alekseyev, Jan 26 2026
LINKS
Max Alekseyev, Table of n, a(n) for n = 1..34 (terms 1..29 from Hiroaki Yamanouchi, 30..31 from Alexander Violette)
EXAMPLE
The abundance of 22 = (1+2+11+22)-44 = -8
MATHEMATICA
Select[Range[10^6], DivisorSigma[1, #] - 2 # == -8 &] (* Michael De Vlieger, Jul 21 2016 *)
PROG
(PARI) for(n=1, 1000000, if(((sigma(n)-2*n)==-8), print1(n, ", ")))
(Magma) [n: n in [1..2*10^7] | (DivisorSigma(1, n)-2*n) eq - 8]; // Vincenzo Librandi, Jul 22 2016
CROSSREFS
KEYWORD
nonn
AUTHOR
Jason G. Wurtzel, Nov 25 2006
EXTENSIONS
a(13)-a(15) from Klaus Brockhaus, Nov 29 2006
a(16)-a(17) from Donovan Johnson, Dec 23 2008
a(18) from Donovan Johnson, Dec 08 2011
a(19) from Giovanni Resta, Mar 29 2013
a(20)-a(25) from Hiroaki Yamanouchi, Aug 21 2018
STATUS
approved