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A259231 Primitive numbers whose abundance is odd. 2

%I

%S 18,100,196,968,1352,2450,4624,5776,6050,8450,8464,11025,13456,15376,

%T 43808,53792,59168,70688,81796,89888,111392,119072,139876,174724,

%U 195364,245025,256036,287296,322624,341056,342225,399424,440896,506944,602176,652864,678976,732736,760384,817216,834632,1032256

%N Primitive numbers whose abundance is odd.

%C A proper subset of A156903.

%C From _Sergey Pavlov_, Mar 22 2017: (Start)

%C Conjecture: let m == a(n) mod 2. Then a(n) can be written as (2+m)^t * d^2 where t is integer, t > 0, d is odd, d > 1.

%C In other words, while a(n) is even, it can be written as 2^t * d^2; while a(n) is odd, it can be written as 3^t * d^2.

%C (Note: for 0 < n < 450, while a(n) is odd, in most cases it is divisible by 5 and in all such cases a(n) can be written as 3^2 * d^2 where d == 0 (mod 5). The only four exceptions are: a(222) = 81162081 = 3^4 * 1001^2; a(255) = 138791961 = 3^4 * 1309^2; a(273) = 173369889 = 3^4 * 1463^2; a(379) = 441882441 = 3^2 * 7007^2.)

%C (End)

%H Giovanni Resta, <a href="/A259231/b259231.txt">Table of n, a(n) for n = 1..5068</a> (terms < 10^12, first 449 from Robert G. Wilson v)

%e 18, a(1), is in the sequence, but none of its multiples are.

%e The first nonmultiple of 18 in A156903 is 100, so it is a(2).

%t L = {}; Do[ab = DivisorSigma[1, n] - 2 n; If[ab > 0 && OddQ[ab] && ! Or @@ (IntegerQ /@ (n/L)), AppendTo[L, n]], {n, 10^5}]; L (* _Giovanni Resta_, Mar 25 2017 *)

%o (PARI) isoddab(n) = my(ab=sigma(n)-2*n); (ab > 0) && (ab % 2);

%o isok(n) = if (isoddab(n), fordiv(n, d, if ((d!=n) && isoddab(d), return (0))); return (1);); \\ _Michel Marcus_, Mar 24 2017

%Y Cf. A156903.

%K nonn

%O 1,1

%A _Robert G. Wilson v_, Jun 21 2015

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Last modified September 25 15:27 EDT 2021. Contains 347658 sequences. (Running on oeis4.)