

A125246


Numbers m whose abundance sigma(m)  2m = 4. Numbers whose deficiency is 4.


16



5, 14, 44, 110, 152, 884, 2144, 8384, 18632, 116624, 8394752, 15370304, 73995392, 536920064, 2147581952, 34360131584, 27034175140420610, 36028797421617152, 576460753914036224
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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^(k1)*(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 (m2)/2 is a term of A067680.  Jinyuan Wang, Apr 08 2020


LINKS

Table of n, a(n) for n=1..19.


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^(k1)*p) = (2^k1)*(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

Cf. A033880, A057732, A067680, A191363, A141548, A125247, A125248, A088832 (abundance 4).
Sequence in context: A222988 A327594 A034530 * A302762 A140796 A197212
Adjacent sequences: A125243 A125244 A125245 * A125247 A125248 A125249


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



