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%I #37 Aug 28 2018 10:41:12
%S 3,15,68,232,543232,2139136,8796271280128,35184745381888
%N Numbers equal to the sum of their aliquot parts, each of them increased by 2.
%C Searched up to n = 10^12.
%C a(9) > 10^18. - _Hiroaki Yamanouchi_, Aug 28 2018
%C From _Giovanni Resta_, May 11 2018: (Start)
%C If p = 2^(1+t) + (1+2*t)*k - 1 is a prime, for some t > 0 and k even, then x = 2^t*p is in the sequence where k is the value by which the sum of aliquot parts is increased.
%C In this sequence k = 2; for t = 21 we get 8796271280128 which is a term greater than 2139136, but this does not exclude the existence of other intermediate terms following a different solution pattern.
%C (End)
%C Terms using odd values of k seem very hard to find. Up to n = 10^12, only three such terms are known: 2, 98, and 8450, for k = 1, 5, and -7, respectively.
%e Aliquot part of 3 is 1 and 1+2 = 3.
%e Aliquot parts of 15 are 1, 3, 5 and (1+2) + (3+2) + (5+2) = 15.
%p with(numtheory): P:=proc(q,k) local n;
%p for n from 1 to q do if 2*n=sigma(n)+k*(tau(n)-1) then print(n);
%p fi; od; end: P(10^12,2);
%t Select[Range[10^6], DivisorSum[#, # + 2 &] - (# + 2) == # &] (* _Michael De Vlieger_, May 14 2018 *)
%Y Cf. A000005, A000203, A000396, A304277, A304278, A304279, A304280, A304281, A304282, A304283, A304284.
%K nonn,hard,more
%O 1,1
%A _Paolo P. Lava_, _Giovanni Resta_, May 11 2018
%E a(7)-a(8) from _Hiroaki Yamanouchi_, Aug 28 2018
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