OFFSET
1,1
COMMENTS
Primitive terms of A156903.
See A156903 for the properties of those numbers. Pavlov's 2017 comment below makes observations of some interest, but its conjecture is false -- see Amiram Eldar's comment in A156942 for the first of its terms that is not divisible by 3. - Peter Munn, Oct 01 2025
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)
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
KEYWORD
nonn
AUTHOR
Robert G. Wilson v, Jun 21 2015
EXTENSIONS
Edited by Peter Munn, Oct 01 2025
STATUS
approved