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A141549 Numbers k whose deficiency is 12: 2k - sigma(k) = 12.

%I A141549 #50 Jun 11 2025 12:33:31
%S A141549 13,45,76,688,8896,133888,537051136,35184418226176,144115191028645888,
%T A141549 2305843021024854016
%N A141549 Numbers k whose deficiency is 12: 2k - sigma(k) = 12.
%C A141549 Numbers n whose abundance is -12. No other terms up to n=100,000,000. - _Jason G. Wurtzel_, Aug 24 2010
%C A141549 For all k in A102633, the number 2^(k-1)*(2^k+11) is in this sequence. So far all terms except a(2) are of this form. For k = 55, 71, this yields terms 649037107316853651724695645454336, 2787593149816327892704951291908936712585216. - _M. F. Hasler_, Apr 23 2015; edited by _Max Alekseyev_, May 27 2025
%C A141549 Any term x = a(m) can be combined with any term y = A141545(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 not produced an amicable pair. However, if one is ever found, then it will exhibit x-y = 12. - _Timothy L. Tiffin_, Sep 13 2016
%C A141549 a(11) > 10^20. - _Max Alekseyev_, May 27 2025
%e A141549 a(1) = 13, since 2*13 - sigma(13) = 26 - 14 = 12. - _Timothy L. Tiffin_, Sep 13 2016
%t A141549 lst={};Do[If[n==Plus@@Divisors[n]-n+12,AppendTo[lst,n]],{n,10^4}];Print[lst];
%t A141549 Select[Range[1, 10^8], DivisorSigma[1, #] - 2 # == - 12 &] (* _Vincenzo Librandi_, Sep 14 2016 *)
%o A141549 (PARI) for(n=1, 10^8, if(((sigma(n)-2*n)==-12), print1(n, ", "))) \\ _Jason G. Wurtzel_, Aug 24 2010
%o A141549 (Magma) [n: n in [1..9*10^6] | (SumOfDivisors(n)-2*n) eq -12]; // _Vincenzo Librandi_, Sep 14 2016
%Y A141549 Cf. A000203, A033880, A005100; A191363 (deficiency 2), A125246 (deficiency 4), A141548 (deficiency 6), A125247 (deficiency 8), A101223 (deficiency 10), A141550 (deficiency 14), A125248 (deficiency 16), A223608 (deficiency 18), A223607 (deficiency 20); A141545 (abundance 12).
%K A141549 nonn,hard,more
%O A141549 1,1
%A A141549 _Vladimir Joseph Stephan Orlovsky_, Aug 16 2008
%E A141549 a(7) from _Donovan Johnson_, Dec 08 2011
%E A141549 a(8)-a(9) from _Hiroaki Yamanouchi_, Aug 21 2018
%E A141549 a(10) from _Max Alekseyev_, May 27 2025