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A156903 Abundant numbers whose abundance is odd. 6
18, 36, 72, 100, 144, 162, 196, 200, 288, 324, 392, 400, 450, 576, 648, 784, 800, 882, 900, 968, 1152, 1296, 1352, 1458, 1568, 1600, 1764, 1800, 1936, 2178, 2304, 2450, 2500, 2592, 2704, 2916, 3042, 3136, 3200, 3528, 3600, 3872, 4050, 4356, 4608, 4624 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

Seems to be a proper subset of A083211. - Robert G. Wilson v, Mar 30 2010

From Robert G. Wilson v, Jun 21 2015: (Start)

If n is present, so is 2*n.

The primitive terms are 18, 100, 196, 968, 1352, 2450, 4624, 5776, 6050, 8450, 8464, 11025, ..., . (A259231).

The number of terms < 10^k:  0, 3, 20, 71, 229, 732, 2319, 7301, 22926, ... (End)

If n is in the sequence, so is 100*n (conjectured). - Sergey Pavlov, Mar 22 2017

Pavlov's observation trivially follows from the fact that to have odd abundance a number k must be either a square or twice a square. If such a number k is abundant then 100*k = (10^2) * k is abundant as well and has odd abundance. In general, we can say that if k is present, so are t^2*k and 2*t^2*k, for every t>0. - Giovanni Resta, Oct 16 2018

n is congruent to {0, 2, 4, 8, 9, 14, 16, 18, 20, 26, 28, 32} (mod 36). - Robert G. Wilson v, Dec 09 2018

LINKS

Robert G. Wilson, Table of n, a(n) for n = 1..22927 (corrected by Michel Marcus)

Eric Weisstein's World of Mathematics, Abundant Number

EXAMPLE

k = 18 is in the sequence because its divisors are {1,2,3,6,9,18} which sum to sigma(k) = 39; so its abundance is sigma(k) - 2k = 39 - 36 = 3.

MAPLE

with(numtheory): select(k->sigma(k)>2*k and modp(sigma(k)-2*k, 2)=1, [$1..5000]); # Muniru A Asiru, Dec 11 2018

MATHEMATICA

abundance[n_] := DivisorSigma[1, n] - 2 n; Select[Range[1000], abundance[#] > 0 && Mod[abundance[#], 2] == 1 &]

abundOddAbundQ[n_] := If[MemberQ[{0, 2, 4, 8, 9, 14, 16, 18, 20, 26, 28, 32}, Mod[n, 36]], a = DivisorSigma[1, n]; OddQ@a && a > 2 n]; Select[ Range@ 5000, abundOddAbundQ@# &] (* Robert G. Wilson v, Dec 23 2018 *)

PROG

(PARI) is(n)=my(k=sigma(n)-2*n); k>0 && k%2 \\ Charles R Greathouse IV, Feb 21 2017

(Python)

from sympy.ntheory import divisor_sigma

def a(n): return divisor_sigma(n) - 2*n

print [n for n in xrange (18, 5001) if a(n)>0 and a(n)%2==1 ] # Indranil Ghosh, Mar 22 2017

(GAP) Filtered([1..5000], k->Sigma(k)-2*k>0 and (Sigma(k)-2*k) mod 2=1); # Muniru A Asiru, Dec 11 2018

CROSSREFS

Cf. A005101, A259231. A proper subset of A083211.

Sequence in context: A087967 A070224 A083211 * A204824 A252424 A115550

Adjacent sequences:  A156900 A156901 A156902 * A156904 A156905 A156906

KEYWORD

nonn

AUTHOR

Robert G. Wilson v, Feb 17 2009

EXTENSIONS

Name edited by Michel Marcus and Charles R Greathouse IV, Mar 26 2017

STATUS

approved

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Last modified May 21 18:56 EDT 2019. Contains 323444 sequences. (Running on oeis4.)