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Numbers m whose deficiency is 10, or: sigma(m) = 2m - 10.
10

%I #46 Sep 08 2022 08:45:16

%S 11,21,26,68,656,2336,8768,133376,528896,34360918016,35184409837568,

%T 576460757135261696

%N Numbers m whose deficiency is 10, or: sigma(m) = 2m - 10.

%C a(13) > 10^18. - _Hiroaki Yamanouchi_, Aug 21 2018

%C A subsequence of A274556. a(11) <= b(23) = 35184409837568 ~ 3.5*10^13, since b(k) := 2^(k-1)*(2^k+9) is in this sequence for all k in A057196 (2^k+9 is prime). All known terms except a(2) = 21 are of that form. - _M. F. Hasler_, Jul 18 2016

%C Any term x of this sequence can be combined with any term y of A223609 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

%e The divisors of 68 are {1, 2, 4, 17, 34, 68} and so sigma(68) = 1 + 2 + 4 + 17+ 24 + 68 = 126 = 2*68 - 10; thus, the deficiency of 68 is 10 so 68 is a term of the sequence.

%t Select[ Range[ 85000000], DivisorSigma[1, # ] + 10 == 2# &]

%o (Magma) [n: n in [1..9*10^6] | (SumOfDivisors(n)) eq 2*n-10]; // _Vincenzo Librandi_, Sep 15 2016

%Y Cf. A033879, A033880, A125246 (deficiency 4), A141548 (deficiency 6), A125247 (deficiency 8), A125248 (deficiency 16).

%Y Cf. also A274556.

%Y Cf. A223609 (abundance 10).

%K nonn,more

%O 1,1

%A Vassil K. Tintschev (tinchev(AT)sunhe.jinr.ru), Dec 15 2004

%E Edited and extended by _Robert G. Wilson v_, Dec 15 2004

%E a(10) from _Donovan Johnson_, Dec 23 2008

%E Edited by _M. F. Hasler_, Jul 18 2016

%E a(11)-a(12) from _Hiroaki Yamanouchi_, Aug 21 2018