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%I #21 May 19 2018 17:42:04
%S 120,460,1472,57584,69488,76516,93148,231748,600928,1924096,8009728,
%T 8043652,33626692,1078034816,2139324416,2535523012,8572567552,
%U 188403300352
%N Numbers equal to the sum of their aliquot parts, each of them decreased by 8.
%C Searched up to n = 10^12.
%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 = -8; for t = 23 we get 140734325850112, which is a term greater than 188403300352, but this does not exclude the existence of other intermediate terms following a different solution pattern.
%C In fact, there could be also sporadic solutions of the type x = 2^t*r*q, where r and q are prime and for which no closed form is known. E.g. for k = -8 we have x = 2^26*134442677*80216006459.
%C To find them, since d(n) = 4*(t+1) and sigma(n) = (2^(t+1)-1)*(1+r)*(1+q), the relation 2*n = sigma(n) + k*(d(n)-1) becomes 2^(t+1)*r*q = (2^(t+1)-1)*(1+r)*(1+q) + k*(4*t+3), which, for fixed t and k, is a quadratic Diophantine equation in r and q that could admit solutions with r and q prime.
%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 parts of 120 are 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 40, 60 and (1-8) + (2-8) + (3-8) + (4-8) + (5-8) + (6-8) + (8-8) + (10-8) + (12-8) + (15-8) + (20-8) + (24-8) + (30-8) + (40-8) + (60-8) = 120.
%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,-8);
%t With[{k = -8}, Select[Range[10^6], DivisorSum[#, # + k &] - (# + k) == # &] ] (* _Michael De Vlieger_, May 14 2018 *)
%Y Cf. A000005, A000203, A000396, A304276, A304277, A304278, A304279, A304280, A304281, A304282, A304283.
%K nonn,hard,more
%O 1,1
%A _Paolo P. Lava_, _Giovanni Resta_, May 11 2018
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