A171929 - OEIS

%I #54 Feb 16 2025 08:33:11

%S 1,3,9,15,45,105,315,1155,7425,8415,8925,31815,32445,351351,442365,

%T 13800465,14571585,16286445,20355825,20487159,78524145,132701205,

%U 159030135,815634435,2586415095,29169504045,40833636525,125208115065

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

%C 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 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 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 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 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 Of the initial 40 terms, only term 45 is in A228058 (and also in A228059). - _Antti Karttunen_, Jan 04 2025

%H 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 Mathworld, <a href="https://mathworld.wolfram.com/Abundancy.html">Abundancy</a>.

%H <a href="/index/O#opnseqs">Index entries for sequences where odd perfect numbers must occur, if they exist at all</a>.

%e Example: a(8) = 1155 since sigma(1155)/1155 = 1.9948 which is closer to 2 than any smaller a(n).

%t 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 (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 Cf. A000203, A000396 (perfect numbers), A053624, A119239, A088012, A117349; A188263 and A188597 (the same but restricted to only abundant resp. deficient numbers).

%Y Cf. also A088012, A228058, A228059.

%K hard,nonn,changed

%O 1,2

%A _Sergio Pimentel_, Jan 05 2010

%E Name improved by _T. D. Noe_, Jan 28 2010

%E More terms from _Max Alekseyev_, _T. D. Noe_ and J. Mulder (jasper.mulder(AT)planet.nl), Jan 26 2010