A223612 - OEIS

%I #35 Sep 11 2023 01:54:37

%S 1312,29824,8341504,134029312,34356723712

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

%C Suggested by _N. J. A. Sloane_ and _Robert G. Wilson v_.

%C a(6) > 10^12.

%C a(6) > 10^13. - _Giovanni Resta_, Mar 29 2013

%C a(6) > 10^18. - _Hiroaki Yamanouchi_, Aug 23 2018

%C 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

%C 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

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

%t Select[Range[1, 10^6], DivisorSigma[1, #] - 2 # == 22 &] (* _Vincenzo Librandi_, Sep 14 2016 *)

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

%o (Magma) [n: n in [1..9*10^6] | (SumOfDivisors(n)-2*n) eq 22]; // _Vincenzo Librandi_, Sep 14 2016

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

%K nonn,more

%O 1,1

%A _Donovan Johnson_, Mar 23 2013

%E Name edited by _Timothy L. Tiffin_, Sep 10 2023