A156903 Abundant numbers (A005101) whose abundance is odd.

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

Offset: 1

Author

Robert G. Wilson v, Feb 17 2009

Keywords

• nonn

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

Examples

			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.
		

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

Crossrefs

Intersection of A005101 and A028982. - Amiram Eldar, Jun 20 2020

Cf. A000203, A033880, A259231.

A proper subset of A083211. Setwise difference A083211 \ A171641.

Positions of odd negative terms in A378600.

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).

Programs

GAP

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

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 *)

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

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