A099477 -id:A099477 - 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

A099475 Number of divisors d of n such that d+2 is also a divisor of n.

Original entry on oeis.org

0, 0, 1, 1, 0, 1, 0, 1, 1, 0, 0, 3, 0, 0, 2, 1, 0, 1, 0, 1, 1, 0, 0, 4, 0, 0, 1, 1, 0, 2, 0, 1, 1, 0, 1, 3, 0, 0, 1, 2, 0, 1, 0, 1, 2, 0, 0, 4, 0, 0, 1, 1, 0, 1, 0, 1, 1, 0, 0, 5, 0, 0, 2, 1, 0, 1, 0, 1, 1, 1, 0, 4, 0, 0, 2, 1, 0, 1, 0, 2, 1, 0, 0, 4, 0, 0, 1, 1, 0, 2, 0, 1, 1, 0, 0, 4, 0, 0, 2, 1, 0, 1, 0, 1, 3

Offset: 1

Reinhard Zumkeller, Oct 18 2004

Number of r X s rectangles with integer sides such that r < s, r + s = 2n, r | s and (s - r) | (s * r). - Wesley Ivan Hurt, Apr 24 2020

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A007862 (similar but with d+1 instead).
Cf. A008585, A008586, A008588.
Cf. A059267, A099476, A099477.

A099475:= proc(n)
local d;
  d:= numtheory:-divisors(n);
nops(d intersect map(`+`,d,2))
end proc:
map(A099475,[$1..1000]); # Robert Israel, Jun 19 2015

a[n_] := DivisorSum[n, Boole[Divisible[n, #+2]]&]; Array[a, 105] (* Jean-François Alcover, Dec 07 2015 *)

A099475(n) = { sumdiv(n, d, ! (n % (d+2))) } \\ Michel Marcus, Jun 18 2015

0 <= a(n) <= a(m*n) for all m>0;
a(A099477(n)) = 0; a(A059267(n)) > 0;
a(A099476(n)) = n and a(m) <> n for m < A099476(n).
For n>0: a(A008585(n))>0, a(A008586(n))>0 and a(A008588(n))>0.
a(n) = Sum_{i=1..n-1} chi((2*n-i)/i) * chi(i*(2*n-i)/(2*n-2*i)), where chi(n) = 1 - ceiling(n) + floor(n). - Wesley Ivan Hurt, Apr 24 2020

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

A108118 Integers not divisible by 3 or 4.

Original entry on oeis.org

1, 2, 5, 7, 10, 11, 13, 14, 17, 19, 22, 23, 25, 26, 29, 31, 34, 35, 37, 38, 41, 43, 46, 47, 49, 50, 53, 55, 58, 59, 61, 62, 65, 67, 70, 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

Offset: 1

Zak Seidov, Jun 04 2005

Or, numbers congruent to {1, 2, 5, 7, 10, 11} mod 12 (cf. A007310). Expand (x+x^2+x^5+x^7+x^10+x^11)/(1-x^12) (cf. A007310). All terms, except 35 and 70, are also in A099477.

Index entries for linear recurrences with constant coefficients, signature (2,-1,-1,2,-1).

Cf. A007310, A099477.

[n : n in [0..150] | n mod 12 in [1, 2, 5, 7, 10, 11]]; // Wesley Ivan Hurt, Jul 22 2016

A108118:=n->12*floor(n/6)+[1, 2, 5, 7, 10, 11][(n mod 6)+1]: seq(A108118(n), n=0..100); # Wesley Ivan Hurt, Jul 22 2016

Select[ Range[132], !IntegerQ[ #/4] && !IntegerQ[ #/3] &] (* or *) Flatten[ NestList[12 + # &, {1, 2, 5, 7, 10, 11}, 10]]

G.f.: x*(1+x^2)^2 / ( (1+x)*(x^2-x+1)*(x-1)^2 ). - R. J. Mathar, Oct 25 2011
From Wesley Ivan Hurt, Jul 22 2016: (Start)
a(n) = 2*a(n-1) - a(n-2) - a(n-3) + 2*a(n-4) - a(n-5) for n>5.
a(n) = a(n-6) + 12 for n>6.
a(n) = (6*n - 3 + cos(n*Pi/3) - cos(n*Pi) - sqrt(3)*sin(n*Pi/3))/3.
a(6k) = 12k-1, a(6k-1) = 12k-2, a(6k-2) = 12k-5, a(6k-3) = 12k-7, a(6k-4) = 12k-10, a(6k-5) = 12k-11. (End)
Sum_{n>=1} (-1)^(n+1)/a(n) = (4-sqrt(3))*Pi/12. - Amiram Eldar, Jan 01 2022

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.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

PARI

Formula

Extensions

A099475 Number of divisors d of n such that d+2 is also a divisor of n.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A243865 Number of twin divisors of n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Formula

A108118 Integers not divisible by 3 or 4.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Maple

Mathematica

Formula