

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.


5



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
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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: + (11/4)*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^21 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

with(numtheory): for n from 1 to 1000 do flag := 1: if n mod 4 = 0 then printf(`%d, `, n):flag := 0 fi: for m from 2 to ceil(sqrt(n)) do if n mod (m^21) = 0 and flag=1 then printf(`%d, `, n); break fi: od: od:


MATHEMATICA

d1d2Q[n_]:=Mod[n, 4]==0AnyTrue[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: A192519 A036446 A284469 * A355200 A049433 A250984
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



