

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
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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
[n for n in range(18, 5001) if a(n) > 0 and a(n) % 2] # 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 A327774
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



