

A125247


Numbers n whose abundance sigma(n)  2n = 8. Numbers n whose deficiency is 8.


16



22, 130, 184, 1012, 2272, 18904, 33664, 70564, 85936, 100804, 391612, 527872, 1090912, 17619844, 2147713024, 6800695312, 34360655872, 549759483904, 1661355408388, 28502765343364, 82994670582016, 99249696661504, 120646991405056, 431202442356004, 952413274955776
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OFFSET

1,1


COMMENTS

a(19) > 10^12.  Donovan Johnson, Dec 08 2011
a(20) > 10^13.  Giovanni Resta, Mar 29 2013
a(30) > 10^18.  Hiroaki Yamanouchi, Aug 21 2018
a(20) <= 36028797958488064 ~ 3.6*10^16. Indeed, if k is in A057195 then 2^(k1)*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
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/41. 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


LINKS

Hiroaki Yamanouchi, Table of n, a(n) for n = 1..29


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

Cf. A033880, A088833 (abundance 8).
Sequence in context: A206418 A309923 A215626 * A249302 A095694 A233060
Adjacent sequences: A125244 A125245 A125246 * A125248 A125249 A125250


KEYWORD

easy,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



