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

945, 1575, 2205, 2835, 3465, 4095, 4725, 5355, 5775, 5985, 6435, 6615, 6825, 7245, 7425, 7875, 8085, 8415, 8505, 8925, 9135, 9555, 9765, 10395, 11025, 11655, 12285, 12705, 12915, 13545, 14175, 14805, 15015, 15435, 16065, 16695, 17325, 17955

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn

Comments

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.

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

From M. F. Hasler, Jul 28 2016: (Start)

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.

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).

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.

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)

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

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

References

• W. Dunham, Euler: The Master of Us All, The Mathematical Association of America Inc., Washington, D.C., 1999, p. 13.
• R. K. Guy, Unsolved Problems in Number Theory, B2.
• James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 128.

Links

• Metin Sariyar, Table of n, a(n) for n = 1..32000 (terms 1..1000 from T. D. Noe)
• Jill Britton, Perfect Number Analyzer.
• L. E. Dickson, Finiteness of the odd perfect and primitive abundant numbers with n distinct prime factors, American Journal of Mathematics 35 (1913), pp. 413-422.
• Mitsuo Kobayashi, Paul Pollack and Carl Pomerance, On the distribution of sociable numbers, Journal of Number Theory, Vol. 129, No. 8 (2009), pp. 1990-2009. See Theorem 10 on p. 2007.
• Victor Meally, Letter to N. J. A. Sloane, no date.
• Walter Nissen, Abundancy : Some Resources.
• Tyler Ross, A Perfect Number Generalization and Some Euclid-Euler Type Results, Journal of Integer Sequences, Vol. 27 (2024), Article 24.7.5. See p. 10.
• Jay L. Schiffman, Odd Abundant Numbers, Mathematical Spectrum, Volume 37, Number 2 (January 2005), pp 73-75.
• Jay L. Schiffman and Christopher S. Simons, More Odd Abundant Sequences, Volume 38, Number 1 (September 2005), pp. 7-8.

Crossrefs

Cf. A000203, A005835, A006038, A115414, A064001, A112640, A122036, A136446, A005101, A173490, A039725, A064989, A337386.

Programs

Maple

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

Mathematica

fQ[n_] := DivisorSigma[1, n] > 2n; Select[1 + 2Range@ 9000, fQ] (* Robert G. Wilson v, Mar 20 2011 *)

PARI

je=[]; forstep(n=1,15000,2, if(sigma(n)>2*n, je=concat(je,n))); je

PARI

is_A005231(n)={bittest(n,0)&&sigma(n)>2*n} \\ M. F. Hasler, Jul 28 2016

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

Formula

a(n) ~ k*n for some constant k (perhaps around 500). - Charles R Greathouse IV, Apr 21 2022

482.8 < k < 489.8 (based on density bounds by Kobayashi et al., 2009). - Amiram Eldar, Jul 17 2022

Extensions

More terms from James Sellers