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A005231
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Odd abundant numbers (odd numbers n whose sum of divisors exceeds 2n).
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48
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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
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OFFSET
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1,1
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COMMENTS
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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 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)
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REFERENCES
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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.
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LINKS
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T. D. Noe, Table of n, a(n) for n=1..1000
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.
Victor Meally, Letter to N. J. A. Sloane, no date.
Walter Nissen, Abundancy : Some Resources
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.
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MAPLE
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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
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MATHEMATICA
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fQ[n_] := DivisorSigma[1, n] > 2n; Select[1 + 2Range@ 9000, fQ] (* Robert G. Wilson v, Mar 20 2011 *)
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PROG
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(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
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CROSSREFS
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Cf. A005835, A006038, A115414, A064001, A112640, A122036, A136446, A005101, A173490, A039725.
Sequence in context: A294576 A252192 A252185 * A174865 A174535 A243104
Adjacent sequences: A005228 A005229 A005230 * A005232 A005233 A005234
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KEYWORD
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nonn
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AUTHOR
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N. J. A. Sloane
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EXTENSIONS
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More terms from James A. Sellers
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STATUS
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approved
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