A059267 Numbers n with 2 divisors d1 and d2 having difference 2: d2 - d1 = 2; equivalently, numbers that are 0 (mod 4) or have a divisor d of the form d = m^2 - 1.

3, 4, 6, 8, 9, 12, 15, 16, 18, 20, 21, 24, 27, 28, 30, 32, 33, 35, 36, 39, 40, 42, 44, 45, 48, 51, 52, 54, 56, 57, 60, 63, 64, 66, 68, 69, 70, 72, 75, 76, 78, 80, 81, 84, 87, 88, 90, 92, 93, 96, 99, 100, 102, 104, 105, 108, 111, 112, 114, 116, 117, 120, 123, 124, 126, 128

Offset: 1

Author

Avi Peretz (njk(AT)netvision.net.il), Jan 23 2001

Keywords

• nonn

Comments

Complement of A099477; A008586, A008585 and A037074 are subsequences - Reinhard Zumkeller, Oct 18 2004

These numbers have an asymptotic density of ~ 0.522. This corresponds to all numbers which are multiples of 4 (25%), or of 3 (having 1 & 3 as divisors: + (1-1/4)*1/3 = 1/4), or of 5*7, or of 11*13, etc. (Generally, multiples of lcm(k,k+2), but multiples of 3 and 4 are already taken into account in the 50% covered by the first 2 terms.) - M. F. Hasler, Jun 02 2012

By considering divisors of the form m^2-1 with m <= 200 it is possible to prove that the density of this sequence is in the interval (0.5218, 0.5226). The numbers of terms not exceeding 10^k, for k = 1, 2, ..., are 5, 52, 521, 5219, 52206, 522146, 5221524, 52215473, 522155386, 5221555813, ..., so the asymptotic density of this sequence can be estimated empirically by 0.522155... . - Amiram Eldar, Sep 25 2022

Examples

			a(18) = 35 because 5 and 7 divide 35 and 7 - 5 = 2.
		

Links

• M. F. Hasler, Table of n, a(n) for n = 1..3131

Crossrefs

Cf. A099475, A099477, A008585, A008586, A037074.

Programs

Maple

isA059267 := proc(n)
    local m ;
    if modp(n,4)=0 then
        true;
    else
        for m from 2 to ceil(sqrt(n)) do
            if modp(n,m^2-1) = 0 then
                return true ;
            end if;
        end do;
        false ;
    end if;
end proc:
for n from 1 to 130 do
    if isA059267(n) then
        printf("%d,",n) ;
    end if;
end do:

Mathematica

d1d2Q[n_]:=Mod[n,4]==0||AnyTrue[Sqrt[#+1]&/@Divisors[n],IntegerQ]; Select[ Range[ 200],d1d2Q] (* Harvey P. Dale, May 31 2020 *)

PARI

isA059267(n)={ n%4==0 || fordiv( n,d, issquare(d+1) && return(1))} \\ M. F. Hasler, Aug 29 2008

PARI

is_A059267(n) = fordiv( n,d, n%(d+2)||return(1)) \\ M. F. Hasler, Jun 02 2012

Formula

A099475(a(n)) > 0. - Reinhard Zumkeller, Oct 18 2004

Extensions

More terms from James Sellers, Jan 24 2001

Removed comments linking to A143714, which seem wrong, as observed by Ignat Soroko, M. F. Hasler, Jun 02 2012