A119239 -id:A119239 - cp's OEIS frontend

A119240 Least odd number k such that sigma(k)/k >= n.

1, 945, 1018976683725, 1853070540093840001956842537745897243375

Offset: 1

T. D. Noe, May 09 2006

nonn

These numbers are a subset of the oddly superabundant numbers, A119239. Laatsch mentions a(3). Pettigrew computes a(4) and a(5), the latter being a 123-digit number.

Pettigrew (link, Tableau 5, p. 21) gives a(5) as 3^6*5^4*7^3*11^2*13^2*17^2*19*...*277. - Jeppe Stig Nielsen, Jul 03 2017

Jeppe Stig Nielsen, Table of n, a(n) for n = 1..5
Richard Laatsch, Measuring the abundancy of integers, Mathematics Magazine 59 (2) (1986) 84-92.
Walter Nissen, Abundancy : Some Resources
Steve Pettigrew, Sur la distribution de nombres spéciaux consécutifs, M.Sc. Thesis, Univ. Laval, 2000.

Cf. A023199 (least number k such that sigma(k)/k >= n).

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

Original entry on oeis.org

1, 3, 9, 15, 45, 105, 315, 1155, 7425, 8415, 8925, 31815, 32445, 351351, 442365, 13800465, 14571585, 16286445, 20355825, 20487159, 78524145, 132701205, 159030135, 815634435, 2586415095, 29169504045, 40833636525, 125208115065

Offset: 1

Sergio Pimentel, Jan 05 2010

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]
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).
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
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
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
Of the initial 40 terms, only term 45 is in A228058 (and also in A228059). - Antti Karttunen, Jan 04 2025

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

Giovanni Resta, Table of n, a(n) for n = 1..40 (terms < 10^13; first 36 terms from T. D. Noe)
Mathworld, Abundancy.
Index entries for sequences where odd perfect numbers must occur, if they exist at all.

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

Cf. also A088012, A228058, A228059.

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 *)

m=2; forstep(n=1,10^10,2, t=abs(sigma(n)/n - 2); if(tMax Alekseyev, Jan 26 2010

Name improved by T. D. Noe, Jan 28 2010

More terms from Max Alekseyev, T. D. Noe and J. Mulder (jasper.mulder(AT)planet.nl), Jan 26 2010

A188263 Odd abundant numbers whose abundancy is closer to 2 than any smaller odd abundant number.

Original entry on oeis.org

945, 2205, 7425, 8415, 8925, 31815, 32445, 351351, 442365, 14571585, 20355825, 20487159, 78524145, 159030135, 1756753845, 2586415095, 82014476355, 93128205975, 125208115065, 127595519865, 154063853475, 394247024535, 948907364895

Offset: 1

T. D. Noe, Mar 30 2011

nonn

The abundancy of a number n is defined as sigma(n)/n. Abundant numbers have an abundancy greater than 2. All these numbers must be odd primitive abundant numbers, A006038.
These numbers might be considered the opposite of A119239, which has odd numbers whose abundancy increases. This sequence has terms in common with A171929. A similar sequence for deficient numbers is A188597.
These are odd numbers that are barely abundant. See A071927 for the even version.
a(24) > 10^12. - Donovan Johnson, May 05 2012

Giovanni Resta, Table of n, a(n) for n = 1..31 (terms < 10^13)

Cf. A171929 (odd numbers whose abundancy is closer to 2 than any smaller odd number)

k = 1; minDiff = 1; Table[k = k + 2; While[abun = DivisorSigma[1, k]/k; abun - 2 > minDiff || abun < 2, k = k + 2]; minDiff = abun - 2; k, {10}]

a(15)-a(16) from Donovan Johnson, Mar 31 2011
a(17)-a(22) from Donovan Johnson, Apr 02 2011
a(23) from Donovan Johnson, May 05 2012

A386423 Odd numbers k such that k/(1+A347381(k)) obtains record values, where A347381 gives the distance from n to the nearest common ancestor of n and sigma(n) in the Doudna-tree.

Original entry on oeis.org

1, 3, 5, 9, 13, 15, 35, 63, 77, 81, 99, 105, 135, 175, 189, 455, 765, 775, 819, 945, 2125, 6375, 9261, 21275, 43011, 43125, 43475, 44469, 45441, 45617, 45619, 46189, 46305, 155363, 161257, 203203, 318835, 401625, 1016015, 1128799, 1773827, 3048045, 3255075, 3386397, 4044555

Offset: 1

Antti Karttunen, Jul 21 2025

Odd terms of A347391 probably form a subsequence, especially if there are no odd perfect numbers or other odd terms larger than one in A336702.

Index entries for sequences where odd perfect numbers must occur, if they exist at all

Cf. A000203, A005940, A336702, A347381, A347391.

Cf. also A119239, A171929, A228059.

m=-1; n=-1; k=0; while(m!=0, n+=2; if(!((n-1)%(2^25)),print1("("n")")); my(r=n/(1+A347381(n))); if(r>m, m=r; k++; write("b386423.txt", k, " ", n); print1(n, ", ")));

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A119240 Least odd number k such that sigma(k)/k >= n.

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

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A188263 Odd abundant numbers whose abundancy is closer to 2 than any smaller odd abundant number.

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A386423 Odd numbers k such that k/(1+A347381(k)) obtains record values, where A347381 gives the distance from n to the nearest common ancestor of n and sigma(n) in the Doudna-tree.

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