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A259231 Primitive numbers whose abundance is odd. 2
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

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 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)

LINKS

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

EXAMPLE

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).

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 *)

PROG

(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

CROSSREFS

Cf. A156903.

Sequence in context: A263999 A087638 A231144 * A064604 A301542 A231138

Adjacent sequences:  A259228 A259229 A259230 * A259232 A259233 A259234

KEYWORD

nonn

AUTHOR

Robert G. Wilson v, Jun 21 2015

STATUS

approved

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Last modified July 31 22:39 EDT 2021. Contains 346377 sequences. (Running on oeis4.)