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A188597 Odd deficient numbers whose abundancy is closer to 2 than any smaller odd deficient number.

%I A188597 #38 Sep 11 2022 16:24:02
%S A188597 1,3,9,15,45,105,315,1155,26325,33705,449295,1805475,10240425,
%T A188597 13800465,16029405,16286445,21003885,32062485,132701205,594397485,
%U A188597 815634435,29169504045,40833636525,295612416135,636988686495,660733931655,724387847085,740099543085,1707894294975,4439852974095,7454198513685
%N A188597 Odd deficient numbers whose abundancy is closer to 2 than any smaller odd deficient number.
%C A188597 The abundancy of a number k is defined as A(k) = sigma(k)/k. Deficient numbers have an abundancy less than 2. This sequence has terms in common with A171929. Sequence A188263, which deals with abundant numbers, approaches 2 from above. The similar sequence for even numbers consists of the powers of 2.
%C A188597 a(29) > 10^12. - _Donovan Johnson_, Apr 08 2011
%C A188597 This sequence is finite iff there is an odd perfect number (which would have abundancy 2). Otherwise, one always has a subsequent term a(n+1) <= a(n)*p where p is the smallest prime not dividing a(n) and larger than 1/(2/A(a(n))-1). Indeed, such an a(n)*p is still deficient but has abundancy larger than a(n), thus closer to 2. - _M. F. Hasler_, Feb 22 2017
%C A188597 From _M. F. Hasler_, Jan 25 2020: (Start)
%C A188597 The upper bounds a(n)*p mentioned above are often terms of the sequence, but not the subsequent but a later one: e.g., 9*5 = 45, 15*7 = 105, 45*7 = 315, 105*11 = 1155, 315*107 = 33705, 1155*389 = 449295, 26325*389 = 10240425, ...
%C A188597 Is 9 the largest term not divisible by 15? Only 7 of the 26 terms listed after 45 are not multiples of 7: is this subsequence finite? (End)
%H A188597 Giovanni Resta, <a href="/A188597/b188597.txt">Table of n, a(n) for n = 1..31</a> (terms < 10^13)
%t A188597 k = 1; minDiff = 1; Join[{k}, Table[k = k + 2; While[abun = DivisorSigma[1, k]/k; 2 - abun > minDiff || abun => 2, k = k + 2]; minDiff = 2 - abun; k, {10}]]
%Y A188597 Cf. A171929 (odd numbers whose abundancy is closer to 2 than any smaller odd number).
%K A188597 nonn
%O A188597 1,2
%A A188597 _T. D. Noe_, Apr 05 2011
%E A188597 a(22)-a(28) from _Donovan Johnson_, Apr 08 2011