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

1,1

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

Links

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

Formula

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

Example

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

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 *)

Prog

(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

Crossrefs

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

Sequence in context: A036446 A284469 A366288 * A362670 A355200 A049433

Adjacent sequences: A059264 A059265 A059266 * A059268 A059269 A059270

Keyword

nonn

Author

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

Extensions

More terms from James A. Sellers, Jan 24 2001

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

Status

approved