A059267 -id:A059267 - cp's OEIS frontend

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

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

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

A362670 Integer inradii for which there exists an isosceles triangle with integer sides (a, a, c) where a < c.

Original entry on oeis.org

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, 129, 132, 135

Offset: 1

Bernard Schott, May 05 2023

nonn

The inradius for isosceles triangle (a, a, c) is r = (c/2)*sqrt((2*a-c)/(2*a+c)).
If m is a term, so is k*m with k > 1.
As r = 3 and r = 4 are terms, A008585 and A008586 are respective subsequences; the only terms < 100 that are not multiples of 3 or 4 are 35 and 70, the next one is r = 154 = 2*7*11 for triple (765, 765, 1386).
By the triangle inequality, a+1 <= c <= 2*a-1.
Differs from A059267. Examples: 154 is not in A059267 but in this sequence at radius r=154 with side lengths c=1386 and a=765. 442 is not in A059267 but in this sequences with r=442, c=6630, a=3435. - R. J. Mathar, Jun 26 2023

			The smallest inradius r = 3 corresponds to isosceles triangle (10, 10, 12).
The second inradius r = 4 corresponds to isosceles triangle (15, 15, 24).
r = 15 is the first inradius for which there exist two such isosceles triangles: (50, 50, 60) and (68, 68, 120).
r = 35 is the smallest inradius that is not multiple of 3 or of 4, this inradius corresponds to isosceles triangle (222, 222, 420).

R. J. Mathar, Solution strategy and Maple program
Eric Weisstein's World of Mathematics, Incircle.
Eric Weisstein's World of Mathematics, Isosceles Triangle.

Cf. A008585, A008586, A070204, A120062, A120570.

Cf. A362669 (similar but with (a,b,b)).

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

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

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A243865 Number of twin divisors of n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Formula

A362670 Integer inradii for which there exists an isosceles triangle with integer sides (a, a, c) where a < c.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

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

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula