OFFSET
1,1
COMMENTS
a(17) > 10^12. - Donovan Johnson, Dec 08 2011
a(17) > 10^13. - Giovanni Resta, Mar 29 2013
a(17) <= b(28) = 36028797421617152 ~ 3.6*10^16, since b(k) := 2^(k-1)*(2^k+3) is in this sequence for all k in A057732, i.e., whenever 2^k+3 is prime, and 28 = A057732(11). Further terms of this form are b(30), b(55), b(67), b(84), ... The only terms not of the form b(k), below 10^13, are {110, 884, 18632, 116624, 15370304, 73995392}. - M. F. Hasler, Apr 27 2015, edited on Jul 17 2016
See A191363 for numbers with deficiency 2, and A141548 for numbers with deficiency 6. - M. F. Hasler, Jun 29 2016 and Jul 17 2016
A term of this sequence multiplied with a prime p not dividing it is abundant if and only if p < sigma(a(n))/4. For each of a(2..16) there is such a prime, near this limit, such that a(n)*p is a primitive weird number, cf. A002975. - M. F. Hasler, Jul 17 2016
Any term x of this sequence can be combined with any term y of A088832 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 5 the only odd number in this sequence? Is it possible to prove this? - M. F. Hasler, Feb 22 2017
a(20) > 10^18. - Hiroaki Yamanouchi, Aug 21 2018
If m is an even term, then (m-2)/2 is a term of A067680. - Jinyuan Wang, Apr 08 2020
EXAMPLE
The abundance of 5 = (1+5)-10 = -4.
More generally, whenever p = 2^k + 3 is prime (as p = 5 for k = 1), then A(2^(k-1)*p) = (2^k-1)*(p+1) - 2^k*p = 2^k - p - 1 = -4.
MATHEMATICA
Select[Range[10^7], DivisorSigma[1, #] - 2 # == -4 &] (* Michael De Vlieger, Jul 18 2016 *)
PROG
(PARI) for(n=1, 1000000, if(((sigma(n)-2*n)==-4), print1(n, ", ")))
(Magma) [n: n in [1..9*10^6] | (SumOfDivisors(n)-2*n) eq -4]; // Vincenzo Librandi, Sep 15 2016
CROSSREFS
KEYWORD
nonn,more
AUTHOR
Jason G. Wurtzel, Nov 25 2006
EXTENSIONS
a(11) to a(14) from Klaus Brockhaus, Nov 29 2006
a(15)-a(16) from Donovan Johnson, Dec 23 2008
a(17)-a(19) from Hiroaki Yamanouchi, Aug 21 2018
STATUS
approved