A259231 Primitive numbers whose abundance is odd.

18, 100, 196, 968, 1352, 2450, 4624, 5776, 6050, 8450, 8464, 11025, 13456, 15376, 43808, 53792, 59168, 70688, 81796, 89888, 111392, 119072, 139876, 174724, 195364, 245025, 256036, 287296, 322624, 341056, 342225, 399424, 440896, 506944, 602176, 652864, 678976, 732736, 760384, 817216, 834632, 1032256

Offset: 1

Author

Robert G. Wilson v, Jun 21 2015

Keywords

• nonn

Comments

A proper subset of A156903.

From Sergey Pavlov, Mar 22 2017: (Start)

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

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.

(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.)

(End)

Examples

			18, a(1), is in the sequence, but none of its multiples are.
The first nonmultiple of 18 in A156903 is 100, so it is a(2).
		

Links

• Giovanni Resta, Table of n, a(n) for n = 1..5068 (terms < 10^12, first 449 from Robert G. Wilson v)

Crossrefs

Cf. A156903.

Programs

Mathematica

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 *)

PARI

isoddab(n) = my(ab=sigma(n)-2*n); (ab > 0) && (ab % 2);
isok(n) = if (isoddab(n), fordiv(n, d, if ((d!=n) && isoddab(d), return (0))); return (1);); \\ Michel Marcus, Mar 24 2017