%I #68 Sep 08 2022 08:46:17

%S 80,1184,6464,29312,78975,510464,557192,137431875584,549741658112,

%T 8796036399104,35184258842624,2251798907715584

%N Numbers n whose abundance is 26: sigma(n) - 2n = 26.

%C Any term x = a(m) can be combined with any term y = A275702(n) to satisfy the property (sigma(x)+sigma(y))/(x+y) = 2. Although this property is a necessary condition for two numbers to be amicable, it is not a sufficient one. So far, these two sequences have produced only one amicable pair: (x,y) = (1184,1210) = (a(2),A275702(5)) = (A063990(3),A063990(4)). If more are ever found, then they will also exhibit y-x = 26.

%C Notice that:

%C a(1) = 80 = 5* 16 = (2*4^2-27)*(4^2)

%C a(2) = 1184 = 37* 32 = (4^3-27)*(4^3)/2

%C a(3) = 6464 = 101* 64 = (2*4^3-27)*(4^3)

%C a(4) = 29312 = 229*128 = (4^4-27)*(4^4)/2

%C a(6) = 510464 = 997*512 = (4^5-27)*(4^5)/2.

%C If p = 2*4^k-27 is prime and n = p*(p+27)/2, then it is not hard to show that sigma(n) - 2*n = 26. The values of k in A275767 will guarantee that p is prime (A275749). Similarly, if q = 4^k-27 is prime and n = q*(q+27)/2, then sigma(n) - 2*n = 26. The values of k in A274519 will guarantee that q is prime (A275750). So, the following values will be in this sequence and provide upper bounds for the next eight terms:

%C (2*4^9-27)*(4^9) = 137431875584 >= a(8)

%C (4^10-27)*(4^10)/2 = 549741658112 >= a(9)

%C (4^11-27)*(4^11)/2 = 8796036399104 >= a(10)

%C (2*4^11-27)*(4^11) = 35184258842624 >= a(11)

%C (4^13-27)*(4^13)/2 = 2251798907715584 >= a(12)

%C (4^25-27)*(4^25)/2 = 633825300114099501099609227264 >= a(13)

%C (4^28-27)*(4^28)/2 = 2596148429267412841487728652582912 >= a(14)

%C (4^29-27)*(4^29)/2 = 41538374868278617137133892585652224 >= a(15).

%C a(8) > 10^9. - _Michel Marcus_, Sep 15 2016

%C a(8) > 2*10^9. - _Michel Marcus_, Dec 31 2016

%C a(13) > 10^18. - _Hiroaki Yamanouchi_, Aug 23 2018

%H D. Alpern, <a href="https://www.alpertron.com.ar/ECM.HTM">Factorization using the Elliptic Curve Method</a>.

%e a(1) = 80, since sigma(80)-2*80 = 186-160 = 26.

%e a(2) = 1184, since sigma(1184)-2*1184 = 2394-2368 = 26.

%e a(3) = 6464, since sigma(6464)-2*6464 = 12954-12928 = 26.

%t Select[Range[10^7], DivisorSigma[1, #] - 2 # == 26 &] (* _Vincenzo Librandi_, Sep 16 2016 *)

%o (PARI) isok(n) = sigma(n) - 2*n == 26; \\ _Michel Marcus_, Sep 15 2016

%o (Magma) [n: n in [1..9*10^6] | (SumOfDivisors(n)-2*n) eq 26]; // _Vincenzo Librandi_, Sep 16 2016

%Y Cf. A033880, A063990, A274519, A275702 (deficiency 26), A275749, A275750, A275767.

%Y Cf. A223609 (abundance 10), ..., A223613 (abundance 24).

%K nonn,more

%O 1,1

%A _Timothy L. Tiffin_, Aug 05 2016

%E a(8)-a(12) from _Hiroaki Yamanouchi_, Aug 23 2018