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A275997 Numbers k whose deficiency is 64: 2k - sigma(k) = 64.

%I A275997 #37 Aug 27 2025 18:00:25
%S A275997 134,284,410,632,1292,1628,4064,9752,12224,22712,66992,72944,403988,
%T A275997 556544,2161664,2330528,8517632,13228352,14563832,15422912,20732792,
%U A275997 89472632,134733824,150511232,283551872,537903104,731670272,915473696,1846850576,2149548032,2159587616
%N A275997 Numbers k whose deficiency is 64: 2k - sigma(k) = 64.
%C A275997 Any term x = a(m) in this sequence can be used with any term y in A275996 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.
%C A275997 The smallest amicable pair is (220, 284) = (A275996(2), a(2)) = (A063990(1), A063990(2)), where 284 - 220 = 64 is the abundance of 220 and the deficiency of 284.
%C A275997 The amicable pair (66928, 66992) = (A275996(7), a(11)) = (A063990(18), A063990(19)), where 66992 - 66928 = 64 is the deficiency of 66992 and the abundance of 66928.
%C A275997 Contains numbers 2^(k-1)*(2^k + 63) whenever 2^k + 63 is prime. - _Max Alekseyev_, Aug 27 2025
%H A275997 Max Alekseyev, <a href="/A275997/b275997.txt">Table of n, a(n) for n = 1..89</a>
%e A275997 a(1) = 134, since 2*134 - sigma(134) = 268 - 204 = 64.
%t A275997 Select[Range[10^7], 2 # - DivisorSigma[1, #] == 64 &] (* _Michael De Vlieger_, Jan 10 2017 *)
%o A275997 (PARI) isok(n) = 2*n - sigma(n) == 64; \\ _Michel Marcus_, Dec 30 2016
%Y A275997 Deficiency k: A191363 (k=2), A125246 (k=4), A141548 (k=6), A125247 (k=8), A101223 (k=10), A141549 (k=12), A141550 (k=14), A125248 (k=16), A223608 (k=18), A223607 (k=20), A223606 (k=22), A385255(k=24), A275702 (k=26), A387352 (k=32).
%Y A275997 Abundance k: A088831 (k=2), A088832 (k=4), A087167 (k=6), A088833 (k=8), A223609 (k=10), A141545 (k=12), A141546 (k=14), A141547 (k=16), A223610 (k=18), A223611 (k=20), A223612 (k=22), A223613 (k=24), A275701 (k=26), A175989 (k=32), A275996 (k=64), A292626 (k=128).
%Y A275997 Cf. A002025, A063990.
%K A275997 nonn,changed
%O A275997 1,1
%A A275997 _Timothy L. Tiffin_, Aug 16 2016
%E A275997 a(23)-a(31) from _Jinyuan Wang_, Mar 02 2020