A125247 Numbers n whose abundance sigma(n) - 2n = -8. Numbers n whose deficiency is 8.

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

Offset: 1

Author

Jason G. Wurtzel, Nov 25 2006

Keywords

• easy
• nonn

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^(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

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

Examples

			The abundance of 22 = (1+2+11+22)-44 = -8
		

Links

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

Crossrefs

Cf. A033880, A088833 (abundance 8).

Programs

Magma

[n: n in [1..2*10^7] | (DivisorSigma(1,n)-2*n) eq - 8]; // Vincenzo Librandi, Jul 22 2016

Mathematica

Select[Range[10^6], DivisorSigma[1, #] - 2 # == -8 &] (* Michael De Vlieger, Jul 21 2016 *)

PARI

for(n=1,1000000,if(((sigma(n)-2*n)==-8),print1(n,",")))

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