cp's OEIS Frontend

A171929 Odd numbers whose abundancy is closer to 2 than any smaller odd number.

%I A171929 #54 Feb 16 2025 08:33:11
%S A171929 1,3,9,15,45,105,315,1155,7425,8415,8925,31815,32445,351351,442365,
%T A171929 13800465,14571585,16286445,20355825,20487159,78524145,132701205,
%U A171929 159030135,815634435,2586415095,29169504045,40833636525,125208115065
%N A171929 Odd numbers whose abundancy is closer to 2 than any smaller odd number.
%C A171929 The (relative) abundancy of n is sigma(n)/n, not sigma(n) - 2n. - _M. F. Hasler_, Apr 12 2015 [As far as I know, "abundancy" has only this meaning; the much less useful sigma(n) - 2n is called "abundance". - _Charles R Greathouse IV_, Feb 19 2017]
%C A171929 So far all known perfect numbers (abundancy = 2) are even, cf. A000396 = (6, 28, 496, 8128, ...). It has been conjectured but not proved that there are no odd perfect numbers. This sequence provides the list of odd numbers that approach perfection (odd numbers which abundancy is closer to two than the abundancy of any smaller odd number).
%C A171929 Odd numbers n such that abs(sigma(n)/n-2) < abs(sigma(m)/m-2) for all m < n. That is, each n is closer to being an odd perfect number than the preceding n. Interestingly, if abs(sigma(n)/n-2) is expressed as a reduced fraction, the numerator of the fraction is 2 for 25 out of the first 30 terms. Terms a(29) and a(30) are 127595519865 and 154063853475. - _T. D. Noe_, Jan 28 2010
%C A171929 Indices of successive minima in the sequence |A000203(n)/n - 2| for odd n. The sequence would terminate at the smallest odd perfect number (if it exists). - _Max Alekseyev_, Jan 26 2010
%C A171929 This sequence is finite if and only there is an odd perfect number. "If" is evident. "Only if" follows because for any real number r > 1 there is an odd number m relatively prime to a given integer such that 1 < sigma(m)/m < r. For example, take a large enough prime. - _Charles R Greathouse IV_, Dec 13 2016, corrected Feb 19 2017
%C A171929 Of the initial 40 terms, only term 45 is in A228058 (and also in A228059). - _Antti Karttunen_, Jan 04 2025
%H A171929 Giovanni Resta, <a href="/A171929/b171929.txt">Table of n, a(n) for n = 1..40</a> (terms < 10^13; first 36 terms from T. D. Noe)
%H A171929 Mathworld, <a href="https://mathworld.wolfram.com/Abundancy.html">Abundancy</a>.
%H A171929 <a href="/index/O#opnseqs">Index entries for sequences where odd perfect numbers must occur, if they exist at all</a>.
%e A171929 Example: a(8) = 1155 since sigma(1155)/1155 = 1.9948 which is closer to 2 than any smaller a(n).
%t A171929 minDiff=Infinity; k=-1; Table[k=k+2; While[abun=DivisorSigma[1,k]/k; Abs[2-abun] > minDiff, k=k+2]; minDiff=Abs[2-abun]; k, {15}] (* _T. D. Noe_, Jan 28 2010 *)
%o A171929 (PARI) m=2; forstep(n=1,10^10,2, t=abs(sigma(n)/n - 2); if(t<m,m=t;print1(n,", ");); ); \\ _Max Alekseyev_, Jan 26 2010
%Y A171929 Cf. A000203, A000396 (perfect numbers), A053624, A119239, A088012, A117349; A188263 and A188597 (the same but restricted to only abundant resp. deficient numbers).
%Y A171929 Cf. also A088012, A228058, A228059.
%K A171929 hard,nonn
%O A171929 1,2
%A A171929 _Sergio Pimentel_, Jan 05 2010
%E A171929 Name improved by _T. D. Noe_, Jan 28 2010
%E A171929 More terms from _Max Alekseyev_, _T. D. Noe_ and J. Mulder (jasper.mulder(AT)planet.nl), Jan 26 2010