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A005231 Odd abundant numbers (odd numbers m whose sum of divisors exceeds 2m).

%I A005231 #119 Jul 17 2025 14:46:56
%S A005231 945,1575,2205,2835,3465,4095,4725,5355,5775,5985,6435,6615,6825,7245,
%T A005231 7425,7875,8085,8415,8505,8925,9135,9555,9765,10395,11025,11655,12285,
%U A005231 12705,12915,13545,14175,14805,15015,15435,16065,16695,17325,17955
%N A005231 Odd abundant numbers (odd numbers m whose sum of divisors exceeds 2m).
%C A005231 While the first even abundant number is 12 = 2^2*3, the first odd abundant is 945 = 3^3*5*7, the 232nd abundant number.
%C A005231 Schiffman notes that 945+630k is in this sequence for all k < 52. Most of the first initial terms are of the form. Among the 1996 terms below 10^6, 1164 terms are of that form, and only 26 terms are not divisible by 5, cf. A064001. - _M. F. Hasler_, Jul 16 2016
%C A005231 From _M. F. Hasler_, Jul 28 2016: (Start)
%C A005231 Any multiple of an abundant number is again abundant, see A006038 for primitive terms, i.e., those which are not a multiple of an earlier term.
%C A005231 An odd abundant number must have at least 3 distinct prime factors, and 5 prime factors when counted with multiplicity (A001222), whence a(1) = 3^3*5*7. To see this, write the relative abundancy A(N) = sigma(N)/N = sigma[-1](N) as A(Product p_i^e_i) = Product (p_i-1/p_i^e_i)/(p_i-1) < Product p_i/(p_i-1).
%C A005231 See A115414 for terms not divisible by 3, A064001 for terms not divisible by 5, A112640 for terms coprime to 5*7, and A047802 for other generalizations.
%C A005231 As of today, we don't know of a difference between this set S of odd abundant numbers and the set S' of odd semiperfect numbers: Elements of S' \ S would be perfect (A000396), and elements of S \ S' would be weird (A006037), but no odd weird or perfect number is known. (End)
%C A005231 For any term m in this sequence, A064989(m) is also an abundant number (in A005101), and for any term x in A115414, A064989(x) is in this sequence. If there are no odd perfect numbers, then applying A064989 to these terms and sorting into ascending order gives A337386. - _Antti Karttunen_, Aug 28 2020
%C A005231 There exist infinitely many terms m such that 2*m+1 is also a term. An example of such a term is given by m = 985571808130707987847768908867571007187. - _Max Alekseyev_, Nov 16 2023
%D A005231 W. Dunham, Euler: The Master of Us All, The Mathematical Association of America Inc., Washington, D.C., 1999, p. 13.
%D A005231 R. K. Guy, Unsolved Problems in Number Theory, B2.
%D A005231 James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 128.
%H A005231 Metin Sariyar, <a href="/A005231/b005231.txt">Table of n, a(n) for n = 1..32000</a> (terms 1..1000 from T. D. Noe)
%H A005231 Jill Britton, <a href="http://britton.disted.camosun.bc.ca/perfect/jbperfect.htm">Perfect Number Analyzer</a>.
%H A005231 L. E. Dickson, <a href="http://www.jstor.org/stable/2370405">Finiteness of the odd perfect and primitive abundant numbers with n distinct prime factors</a>, American Journal of Mathematics 35 (1913), pp. 413-422.
%H A005231 Mitsuo Kobayashi, Paul Pollack and Carl Pomerance, <a href="https://doi.org/10.1016/j.jnt.2008.10.011">On the distribution of sociable numbers</a>, Journal of Number Theory, Vol. 129, No. 8 (2009), pp. 1990-2009. See Theorem 10 on p. 2007.
%H A005231 Victor Meally, <a href="/A006556/a006556.pdf">Letter to N. J. A. Sloane</a>, no date.
%H A005231 Walter Nissen, <a href="http://upforthecount.com/math/abundance.html">Abundancy : Some Resources</a>.
%H A005231 Tyler Ross, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL27/Ross/ross3.html">A Perfect Number Generalization and Some Euclid-Euler Type Results</a>, Journal of Integer Sequences, Vol. 27 (2024), Article 24.7.5. See p. 10.
%H A005231 Jay L. Schiffman, <a href="http://www.appliedprobability.org/data/files/MS%20issues/Vol37_No2.pdf">Odd Abundant Numbers</a>, Mathematical Spectrum, Volume 37, Number 2 (January 2005), pp 73-75.
%H A005231 Jay L. Schiffman and Christopher S. Simons, <a href="http://users.rowan.edu/%7Esimons/abundant.pdf">More Odd Abundant Sequences</a>, Volume 38, Number 1 (September 2005), pp. 7-8.
%F A005231 a(n) ~ k*n for some constant k (perhaps around 500). - _Charles R Greathouse IV_, Apr 21 2022
%F A005231 482.8 < k < 489.8 (based on density bounds by Kobayashi et al., 2009). - _Amiram Eldar_, Jul 17 2022
%p A005231 A005231 := proc(n) option remember ; local a ; if n = 1 then 945 ; else for a from procname(n-1)+2 by 2 do if numtheory[sigma](a) > 2*a then return a; end if; end do: end if; end proc: # _R. J. Mathar_, Mar 20 2011
%t A005231 fQ[n_] := DivisorSigma[1, n] > 2n; Select[1 + 2Range@ 9000, fQ] (* _Robert G. Wilson v_, Mar 20 2011 *)
%o A005231 (PARI) je=[]; forstep(n=1,15000,2, if(sigma(n)>2*n, je=concat(je,n))); je
%o A005231 (PARI) is_A005231(n)={bittest(n,0)&&sigma(n)>2*n} \\ _M. F. Hasler_, Jul 28 2016
%o A005231 (PARI) list(lim)=my(v=List()); forfactored(n=945,lim\1, if(n[2][1,1]>2 && sigma(n,-1)>2, listput(v,n[1]))); Vec(v) \\ _Charles R Greathouse IV_, Apr 21 2022
%Y A005231 Cf. A000203, A005835, A006038, A115414, A064001, A112640, A122036, A136446, A005101, A173490, A039725, A064989, A337386.
%K A005231 nonn
%O A005231 1,1
%A A005231 _N. J. A. Sloane_
%E A005231 More terms from _James Sellers_