A171929 -id:A171929 - cp's OEIS frontend

A077374 Odd numbers m whose abundance by absolute value is at most 10, that is, -10 <= sigma(m) - 2m <= 10.

1, 3, 5, 7, 9, 11, 15, 21, 315, 1155, 8925, 32445, 442365, 815634435

Offset: 1

Jason Earls, Nov 30 2002

Apart from {1, 3, 5, 7, 9, 11, 15, 21, 315}, subset of A088012. Probably finite. - Charles R Greathouse IV, Mar 28 2011
a(15) > 10^13. - Giovanni Resta, Mar 29 2013
The abundance of the given terms a(1..14) is: (-1, -2, -4, -6, -5, -10, -6, -10, -6, -6, 6, 6, 6, -6). See also A171929, A188263 and A188597 for numbers with abundancy sigma(n)/n close to 2. - M. F. Hasler, Feb 21 2017
a(15) > 10^22. - Wenjie Fang, Jul 13 2017

			sigma(32445) = 64896 and 32445*2 = 64890, which makes the odd number 32445 six away from perfection: A(32445) = 6 and hence in this sequence.

Eric Weisstein's World of Mathematics, Abundance.

Contains odd terms of: A191363, A088831, A125246, A088832, A141548, A087167, A125247, A088833, A101223, A223609.

Cf. A033879, A033880, A000203, A088012.

Select[Range[1, 10^6, 2], -10 <= DivisorSigma[1, #] - 2 # <= 10 &] (* Michael De Vlieger, Feb 22 2017 *)

forstep(n=1,442365,2,if(abs(sigma(n)-2*n)<=10,print1(n,",")))

a(14) from Farideh Firoozbakht, Jan 12 2004

A088012 Odd solutions to abs(sigma(k) - 2k) <= log(k). Numbers k whose abundance-radius does not exceed log(k).

Original entry on oeis.org

1155, 8925, 32445, 442365, 159030135, 815634435, 2586415095, 1956860570050575, 221753180448460815, 747406020889133775

Offset: 1

Labos Elemer and Farideh Firoozbakht, Oct 20 2003

This sequence should include odd perfect numbers too, if they exist.
From Walter Nissen, Dec 15 2005: (Start)
     abundancy(k)             k            2k       sigma(k)  abundance
   1.99480519480519         1155          2310          2304      -6
   2.00067226890756         8925         17850         17856       6
   2.00018492834027        32445         64890         64896       6
   2.00001356346004       442365        884730        884736       6
   2.00000011318610    159030135     318060270     318060288      18
   1.99999999264376    815634435    1631268870    1631268864      -6
   2.00000000695943   2586415095    5172830190    5172830208      18
As it happens, abundance of these is -6, 6 or 18. This is not necessarily true for larger terms. (End)
See also A171929 and A188597 and A188263 for sequences of numbers (any / deficient / abundant) whose relative abundancy tends to 2. - M. F. Hasler, Feb 19 2017
3278298202600507814120339275775985 is also a term with abundance 30. In fact, it and 815634435 are the only odd terms known where abs(sigma(k)-2k) <= log_10(k). - Alexander Violette, Nov 05 2020; updated by Max Alekseyev, Jul 27 2025
Also includes 827880257692739174385 and 255286886041240176056063754225. - Max Alekseyev, Jul 27 2025

			1155 is in the sequence because sigma(1155) = 2304, giving 2*1155 - 2304 = 6, while natural log of 1155 is about 7.05.
From _M. F. Hasler_, Jul 18 2016: (Start)
We have the following factorizations:
1155 = 3 * 5 * 7 * 11,
8925 = 3 * 5^2 * 7 * 17,
32445 = 3^2 * 5 * 7 * 103,
442365 = 3 * 5 * 7 * 11 * 383,
159030135 = 3^5 * 5 * 11 * 73 * 163,
815634435 = 3 * 5 * 7 * 11 * 547 * 1291,
2586415095 = 3^2 * 5 * 11 * 31 * 41 * 4111.
The sequence appears to be a subsequence of A171929. (End)

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

Cf. A000079, A000396, A005100, A005101, A077374, A087167, A088007-A088011.

Cf. also A171929, A188263, A188597, A228059, A295296.

abu[x_] := Abs[DivisorSigma[1, x]-2*x] Do[If[ !Greater[abu[n], Log[n]//N]&&OddQ[n], Print[n]], {n, 1, 100000}]

is(n)=n%2 && abs(sigma(n)-2*n)<=log(n) \\ Charles R Greathouse IV, Feb 21 2017

a(7) from Donovan Johnson, Dec 21 2008

a(9) from Alexander Violette confirmed and a(8), a(10) added by Max Alekseyev, Jul 27 2025

A228059 Odd numbers of the form p^(1+4k) * r^2, where p is prime of the form 1+4m, r > 1, and gcd(p,r) = 1 that are closer to being perfect than previous terms.

Original entry on oeis.org

45, 405, 2205, 26325, 236925, 1380825, 1660725, 35698725, 3138290325, 29891138805, 73846750725, 194401220013, 194509436121, 194581580193, 194689796301, 194798012409, 194906228517, 194942300553, 195230876841, 195339092949, 195447309057, 195699813309

Offset: 1

T. D. Noe, Aug 14 2013

nonn

A number x is perfect if sigma(x) = 2x, where sigma is the sum of divisors of x. See A228058 for numbers of the form p^(1+4k) * r^2. This sequence ends when the first odd perfect number occurs.
The first two papers by Dris listed below are for information only; this sequence in independent of the papers. In the second paper, Dris attempts to prove that the exponent of p above is 1 for odd perfect numbers. Coincidently, the first 9 numbers in this sequence have exponent 1.
a(38) > 10^12. - Giovanni Resta, Aug 16 2018
a(38) <= 283665529390725 = 15349 * (3^3 * 5 * 19 * 53)^2. - Giovanni Resta, Aug 23 2018
a(39) <= 3116918388785625 = 37993 * (3^2 * 5^2 * 19 * 67)^2. - Alexander Violette, Mar 05 2022
The first 37 terms are all multiples of 3, as well as the two additional terms given above. See also comments in A349752. - Antti Karttunen, Jan 04 2025

			           45 =   5 * 3^2.
          405 =   5 * 3^4.
         2205 =   5 * (3 * 7)^2.
        26325 =  13 * (3^2 * 5)^2.
       236925 =  13 * (3^3 * 5)^2.
      1380825 =  17 * (3 * 5 * 19)^2.
      1660725 =  61 * (3 * 5 * 11)^2.
     35698725 =  61 * (3^2 * 5 * 17)^2.
   3138290325 =  53 * (3^4 * 5 * 19)^2.
  29891138805 =   5 * (3^2 * 11^2 * 71)^2.
  73846750725 = 509 * (3 * 5 * 11 * 73)^2.

Giovanni Resta, Table of n, a(n) for n = 1..37
Jose Arnaldo B. Dris, The abundancy index of divisors of odd perfect numbers, J. Integer Sequences, 15 (2012), Article 12.4.4.
Jose Arnaldo B. Dris, A short "proof" for Sorli's conjecture on odd perfect numbers, arxiv 1308.2156 [math.NT], 2013-2015.
Jose Arnaldo B. Dris, Euclid-Euler Heuristics for (Odd) Perfect Numbers, arXiv preprint arXiv:1310.5616 [math.NT], 2013-2017.
Jose Arnaldo B. Dris, A Sufficient Condition for Disproving Descartes's Conjecture on Odd Perfect Numbers, arXiv preprint arXiv:1311.6803 [math.NT], 2013-2015.
Jose Arnaldo Bebita Dris, Doli-Jane Uvales Tejada, A note on the OEIS sequence A228059, Notes on Number Theory and Discrete Mathematics (2019) Vol. 25, No. 1, 199-205.
Index entries for sequences where any odd perfect numbers must occur

Cf. A000203 (sigma), A000396 (perfect numbers), A228058, A325379, A349752.

Cf. also A171929.

nn = 7; f[n_] := Abs[DivisorSigma[1, n]/n - 2]; n = 45; t = {n}; lastF = f[n]; cnt = 1; While[cnt < nn, n = n + 2; {p, e} = Transpose[FactorInteger[n]]; od = Select[e, OddQ]; If[Length[e] > 1 && Length[od] == 1 && Mod[od[[1]], 4] == 1 && Mod[p[[Position[e, od[[1]]][[1, 1]]]], 4] == 1 && f[n] < lastF, cnt++; lastF = f[n]; Print[{n, lastF}]; AppendTo[t, n]]]; t

isA228058(n) = if(!(n%2)||(omega(n)<2),0,my(f=factor(n),y=0); for(i=1,#f~,if(1==(f[i,2]%4), if((1==y)||(1!=(f[i,1]%4)),return(0),y=1), if(f[i,2]%2, return(0)))); (y));
m=-1; n=0; while(m!=0, n++; if(isA228058(n), if((m<0) || abs((sigma(n)/n)-2)Antti Karttunen, Apr 22 2019

a(10) (as communicated by T. D. Noe) added by Jose Arnaldo Bebita Dris, Aug 16 2018

a(11)-a(22) from Giovanni Resta, Aug 16 2018

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

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

Original entry on oeis.org

1, 3, 9, 15, 45, 105, 315, 1155, 26325, 33705, 449295, 1805475, 10240425, 13800465, 16029405, 16286445, 21003885, 32062485, 132701205, 594397485, 815634435, 29169504045, 40833636525, 295612416135, 636988686495, 660733931655, 724387847085, 740099543085, 1707894294975, 4439852974095, 7454198513685

Offset: 1

T. D. Noe, Apr 05 2011

nonn

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.
a(29) > 10^12. - Donovan Johnson, Apr 08 2011
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
From M. F. Hasler, Jan 25 2020: (Start)
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, ...
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)

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; 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}]]

a(22)-a(28) from Donovan Johnson, Apr 08 2011

A119239 Oddly superabundant numbers: odd n with sigma(n)/n > sigma(k)/k for all odd k < n.

Original entry on oeis.org

1, 3, 9, 15, 45, 105, 315, 945, 1575, 2835, 3465, 10395, 17325, 31185, 45045, 135135, 225225, 405405, 675675, 2027025, 2297295, 3828825, 6891885, 11486475, 34459425, 43648605, 72747675, 130945815, 218243025, 654729075, 1003917915, 1527701175

Offset: 1

T. D. Noe, May 09 2006

nonn

Every oddly colossally abundant number (A110464) is in this sequence.

a(8) = 945 is the first term with abundancy > 2, a(41) = 1018976683725 is the first term with abundancy > 3, and a(141) = 1853070540093840001956842537745897243375 is the first term with abundancy > 4. See A119240. - Antti Karttunen, Jul 21 2025

T. D. Noe, Table of n, a(n) for n = 1..1000
Richard Laatsch, Measuring the abundancy of integers, Mathematics Magazine 59 (2) (1986) 84-92.
Walter Nissen, Abundancy : Some Resources

Cf. A004394 (superabundant numbers), A005231 (odd abundant numbers), A053624 (highly composite odd numbers), A119240.

Cf. also A171929, A228059, A386423.

rec=0; lst={}; Do[abun=DivisorSigma[1,n]/n; If[abun>rec, rec=abun; AppendTo[lst,n]], {n,1,10^6,2}]; lst

r=0;forstep(n=1,1e6,2,t=sigma(n)/n;if(t>r,r=t;print1(n", "))) \\ Charles R Greathouse IV, Nov 27 2013

Definition clarified by Jonathan Sondow, Dec 08 2011

A228450 Deficient numbers with increasing abundancy without being powers of 2.

Original entry on oeis.org

3, 9, 10, 44, 110, 136, 592, 884, 2144, 8384, 18632, 32896, 116624, 391612, 527872, 1090912, 2102272, 8394752, 15370304, 73995392, 536920064, 815634435, 2147516416, 34360131584, 217898810368, 546409576448, 549759483904

Offset: 1

Michel Marcus, Oct 27 2013

Without the additional condition one would have obtained A000079, see "least deficient" comment there. Subsequence of A005100.

			First term is 3 with sigma(n)/n = 4/3 ~ 1.33, then 4 with 13/9 ~ 1.44, then 10 with 9/5 = 1.80.

Cf. A000079, A000203, A005100, A071927, A171929, A188597.

abun[n_] := DivisorSigma[1, n]/n; mx = 0; t = {}; Do[m = abun[n]; If[m < 2 && m > mx && ! IntegerQ[Log[2, n]], mx = m; AppendTo[t, n]], {n, 10000}]; t (* T. D. Noe, Apr 09 2014 *)

lista(nn) = {rab = 0; for (n=1, nn, if (n != 2^valuation(n, 2), ab = sigma(n)/n; if ((ab < 2) && (ab > rab), print1(n, ", "); rab = ab;);););} \\ Michel Marcus, Oct 27 2013

a(21)-a(22) from Michel Marcus, Oct 28 2013

a(23)-a(27) from Donovan Johnson, Nov 13 2013

A386422 Odd numbers k that are closer to being perfect than previous terms and also satisfy the condition that A324644(k)/A324198(k) = 2.

Original entry on oeis.org

3, 33, 99, 135, 855, 2295, 19575, 38745, 63855, 121485, 371925, 3870195, 8109585, 28306005, 36340395, 113215095, 463084245, 672363615, 675916395, 686574735, 1208140395

Offset: 1

Antti Karttunen, Jul 21 2025

Questions: Are there only multiples of 5 after the three initial terms? Are there any common terms with A228058?

Apart from initial 3, a subsequence of A364286.
Cf. A000203, A000396, A228058, A324198, A324644.
Cf. also A171929, A228059, A386419, A386420, A386421 for similar sequences.

A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
is_A364286(n) = if(isprime(n), 0, my(u=A276086(n)); (gcd(sigma(n),u)==2*gcd(n,u))); \\ Antti Karttunen, Jul 21 2025
m=-1; n=-1; k=0; while(m!=0, n+=2; if(!((n-1)%(2^25)),print1("("n")")); if(isprime(n) || is_A364286(n), if((m<0) || abs((sigma(n)/n)-2)

A386419 Odd numbers k that are closer to being perfect than previous terms, and also satisfy the condition that phi(k) = phi(sigma(k)).

Original entry on oeis.org

1, 3, 15, 45, 585, 2295, 11475, 29835, 72675, 424575, 7977165, 28851975, 29277885, 39317175

Offset: 1

Antti Karttunen, Jul 21 2025

Questions: Is 45 the only term also in A228058? (See also A354362). Are there only multiples of 5 after the two initial terms?

If it exists, a(15) > 2^30 (1073741824).

Index entries for sequences where (the least) odd perfect number must occur

Subsequence of A353679.
Cf. A000010, A000203, A000396, A228058, A354362.
Cf. also A171929, A228059, A386420, A386421, A386422.

A353680(n) = ((n%2) && (eulerphi(sigma(n))==eulerphi(n)));
isA353679(n) = A353680(n);
m=-1; n=0; k=0; while(m!=0, n++; if(!(n%(2^25)),print1("("n")")); if(isA353679(n), if((m<0) || abs((sigma(n)/n)-2)

A386420 Odd numbers k that are closer to being perfect than previous terms and also satisfy the conditions that sigma(k) preserves the 3-adic valuation of k, and that sigma(k) == -k (mod 3).

Original entry on oeis.org

7, 15, 105, 495, 1365, 2205, 9405, 26145, 31815, 497835, 654675, 1984455, 7188885, 9018009, 9338595, 9958905, 13777785, 13800465, 14571585, 47020995, 78867495, 132884115, 210124665, 363860775

Offset: 1

Antti Karttunen, Jul 21 2025

Question: Is 2205 the only term also in A228058?

If it exists, a(25) > 1275068416.

Index entries for sequences where (the least) odd perfect number must occur

Subsequence of A349752, thus also of A349749 and of A349751.
Cf. A000203.
Cf. also A171929, A228059, A386419, A386421, A386422.

isA349752(n) = if(!(n%2), 0, my(s=sigma(n)); (0==(s+n)%3) && valuation(s, 3)==valuation(n, 3));
m=-1; n=0; k=0; while(m!=0, n++; if(!(n%(2^25)),print1("("n")")); if(isA349752(n), if((m<0) || abs((sigma(n)/n)-2)

cp's OEIS Frontend

A077374 Odd numbers m whose abundance by absolute value is at most 10, that is, -10 <= sigma(m) - 2m <= 10.

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A088012 Odd solutions to abs(sigma(k) - 2k) <= log(k). Numbers k whose abundance-radius does not exceed log(k).

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A228059 Odd numbers of the form p^(1+4k) * r^2, where p is prime of the form 1+4m, r > 1, and gcd(p,r) = 1 that are closer to being perfect than previous terms.

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

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

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A119239 Oddly superabundant numbers: odd n with sigma(n)/n > sigma(k)/k for all odd k < n.

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A228450 Deficient numbers with increasing abundancy without being powers of 2.

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A386422 Odd numbers k that are closer to being perfect than previous terms and also satisfy the condition that A324644(k)/A324198(k) = 2.