A223612 Numbers k whose abundance is 22: sigma(k) - 2*k = 22.

1312, 29824, 8341504, 134029312, 34356723712

Offset: 1

Author

Donovan Johnson, Mar 23 2013

Keywords

• nonn
• more

Comments

Suggested by N. J. A. Sloane and Robert G. Wilson v.

a(6) > 10^12.

a(6) > 10^13. - Giovanni Resta, Mar 29 2013

a(6) > 10^18. - Hiroaki Yamanouchi, Aug 23 2018

Any term x of this sequence can be combined with any term y of A223606 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. [Proof: If x = a(n) and y = A223606(m), then sigma(x) = 2x+22 and sigma(y) = 2y-22. Thus, sigma(x)+sigma(y) = (2x+22)+(2y-22) = 2x+2y = 2(x+y), which implies that (sigma(x)+sigma(y))/(x+y) = 2(x+y)/(x+y) = 2.] - Timothy L. Tiffin, Sep 13 2016

a(6) <= 2361183240644548624384. Every number of the form 2^(j-1)*(2^j - 23), where 2^j - 23 is prime, is a term. - Jon E. Schoenfield, Jun 02 2019

Examples

			For k = 34356723712, sigma(k) - 2*k = 22.
		

Crossrefs

Cf. A000203, A033880, A223606 (deficiency 22).

Programs

Magma

[n: n in [1..9*10^6] | (SumOfDivisors(n)-2*n) eq 22]; // Vincenzo Librandi, Sep 14 2016

Mathematica

Select[Range[1, 10^6], DivisorSigma[1, #] - 2 # == 22 &] (* Vincenzo Librandi, Sep 14 2016 *)

PARI

for(n=1, 10^8, if(sigma(n)-2*n==22, print1(n ", ")))

Extensions

Name edited by Timothy L. Tiffin, Sep 10 2023