

A156903


Abundant numbers (A005101) whose abundance is odd.


10



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

Equivalently, abundant numbers with odd sum of divisors.
Complement of A204825 with respect to A005101 (abundant numbers).
Seems to be a proper subset of A083211.  Robert G. Wilson v, Mar 30 2010. This sequence is indeed a proper subset of A083211, since the abundance of a number k, A033880(k) = sigma(k)  2*k, has the same parity as sigma(k). If sigma(k) is odd then the sums of any two complementary subsets of the divisors of k have different parities and thus they cannot be equal.  Amiram Eldar, Jun 20 2020
If n is present, so is 2*n.  Robert G. Wilson v, Jun 21 2015
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
Terms are 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 v, 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

Intersection of A005101 and A028982.  Amiram Eldar, Jun 20 2020
Cf. A000203, A033880, A259231. A proper subset of A083211.
Cf. A204825 (abundant numbers with even sum of divisors), A204826 (deficient numbers with odd sum of divisors), A204827 (deficient numbers with even sum of divisors).
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
Edited by N. J. A. Sloane, Jun 21 2020 at the suggestion of Amiram Eldar


STATUS

approved



