A099475 -id:A099475 - cp's OEIS frontend

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

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

nonn

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

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

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

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

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:

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

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

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

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

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

A099477 Numbers having no divisors d such that also d+2 is a divisor.

Original entry on oeis.org

1, 2, 5, 7, 10, 11, 13, 14, 17, 19, 22, 23, 25, 26, 29, 31, 34, 37, 38, 41, 43, 46, 47, 49, 50, 53, 55, 58, 59, 61, 62, 65, 67, 71, 73, 74, 77, 79, 82, 83, 85, 86, 89, 91, 94, 95, 97, 98, 101, 103, 106, 107, 109, 110, 113, 115, 118, 119, 121, 122, 125, 127, 130, 131, 133

Offset: 1

Reinhard Zumkeller, Oct 18 2004

nonn

Except for 3, all primes are in this sequence. - Alonso del Arte, Jun 13 2014

			10 is in the sequence because its divisors are 1, 2, 5, 10, none of which is 2 less than another.
11 is in the sequence as are all primes other than 3.
12 is not in the sequence because its divisors are 1, 2, 3, 4, 6, 12, of which 2 and 4 are 2 less than another divisor.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Complement of A059267.

Cf. A099475, A108118.

twinDivsQ[n_] := Union[ IntegerQ[ # ] & /@ (n/(Divisors[n] + 2))][[ -1]] == True; Select[ Range[133], !twinDivsQ[ # ] &] (* Robert G. Wilson v, Jun 09 2005 *)
d2noQ[n_]:=Module[{d=Divisors[n]},Intersection[d,d+2]=={}]; Select[ Range[ 150],d2noQ] (* Harvey P. Dale, Feb 15 2019 *)

A099475(a(n)) = 0.

A243865 Number of twin divisors of n.

Original entry on oeis.org

0, 0, 2, 2, 0, 2, 0, 2, 2, 0, 0, 5, 0, 0, 3, 2, 0, 2, 0, 2, 2, 0, 0, 6, 0, 0, 2, 2, 0, 3, 0, 2, 2, 0, 2, 5, 0, 0, 2, 4, 0, 2, 0, 2, 3, 0, 0, 6, 0, 0, 2, 2, 0, 2, 0, 2, 2, 0, 0, 8, 0, 0, 4, 2, 0, 2, 0, 2, 2, 2, 0, 6, 0, 0, 3, 2, 0, 2, 0, 4, 2, 0, 0, 7, 0, 0, 2, 2, 0, 3, 0, 2, 2, 0, 0, 6

Offset: 1

Juri-Stepan Gerasimov, Jun 13 2014

nonn

A divisor m of n is a twin divisor if m-2 (for m >= 3) and m+2 (for m <= n-2) also divide n.

			The positive divisors of 20 are 1, 2, 4, 5, 10, 20. Of these, 2 and 4 are twin divisors: (2)+2 = 4, which divides n, and (4)-2 = 2 also divides n. So a(20) = the number of these divisors, which is 2.

Jens Kruse Andersen, Table of n, a(n) for n = 1..10000

Cf. A000005, A002162, A099475, A132747, A243917.

a(n) = sumdiv(n, d, ((d>2) && !(n % (d-2))) || !(n % (d+2))); \\ Michel Marcus, Jun 25 2014

a(n) = A000005(n) - A243917(n).
a(3n) > 1 for all n >= 1.
a(A099477(n)) = 0, a(A059267(n)) > 0.
A099475(n) <= a(n) <= A000005(n).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = log(2)/2 + 17/12 = 1.7632402569... . - Amiram Eldar, Mar 22 2024

A099476 Smallest number having exactly n divisors d such that also d+2 is a divisor.

Original entry on oeis.org

1, 3, 15, 12, 24, 60, 180, 120, 360, 720, 1260, 840, 1680, 3360, 2520, 7560, 5040, 10080, 30240, 32760, 27720, 85680, 83160, 55440, 110880, 166320, 277200, 526680, 360360, 831600, 942480, 1053360, 1801800, 720720, 3769920, 1441440, 2162160

Offset: 0

Reinhard Zumkeller, Oct 18 2004

nonn

Amiram Eldar, Table of n, a(n) for n = 0..87

Cf. A059267, A099475.

f[n_] := DivisorSum[n, Boole@Divisible[n, #+2] &]; m = 20; a = Table[0, {m}]; c = 0; n = 2; While[c < m, f1 = f[n];  If[f1 > 0 && f1 <= m && a[[f1]] == 0, c++; a[[f1]] = n]; n++]; Join[{1}, a] (* Amiram Eldar, Sep 02 2019 *)

A099475(a(n)) = n and A099475(m) <> n for m < a(n).

cp's OEIS Frontend

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.

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A099477 Numbers having no divisors d such that also d+2 is a divisor.

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A243865 Number of twin divisors of n.

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A099476 Smallest number having exactly n divisors d such that also d+2 is a divisor.

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