A066660 -id:A066660 - cp's OEIS frontend

A076984 Number of Fibonacci numbers that are divisors of the n-th Fibonacci number.

1, 1, 2, 2, 2, 3, 2, 3, 3, 3, 2, 5, 2, 3, 4, 4, 2, 5, 2, 5, 4, 3, 2, 7, 3, 3, 4, 5, 2, 7, 2, 5, 4, 3, 4, 8, 2, 3, 4, 7, 2, 7, 2, 5, 6, 3, 2, 9, 3, 5, 4, 5, 2, 7, 4, 7, 4, 3, 2, 11, 2, 3, 6, 6, 4, 7, 2, 5, 4, 7, 2, 11, 2, 3, 6, 5, 4, 7, 2, 9, 5, 3, 2, 11, 4, 3, 4, 7, 2, 11, 4, 5, 4, 3, 4, 11, 2, 5, 6, 8, 2, 7, 2

Offset: 1

Amarnath Murthy, Oct 25 2002

nonn

a(A001605(n)) = 2; a(A105802(n)) = n.

It is well known that if k is a divisor of n then F(k) divides F(n). Hence if n has d divisors, one expects that a(n)=d. However because F(1)=F(2)=1, there is one fewer Fibonacci divisor when n is even. So for even n, a(n)=d-1. - T. D. Noe, Jan 18 2006

			n=12, A000045(12)=144: 5 of the 15 divisors of 144 are also Fibonacci numbers, a(12) = #{1, 2, 3, 8, 144} = 5.

T. D. Noe, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Fibonacci Number

Cf. A000005, A063375, A000045, A105800.
Cf. A076985.
Cf. A068499.

with(combinat, fibonacci):a[1] := 1:for i from 2 to 229 do s := 0:for j from 2 to i do if((fibonacci(i) mod fibonacci(j))=0) then s := s+1:fi:od:a[i] := s:od:seq(a[l],l=2..229);

Table[s=DivisorSigma[0, n]; If[OddQ[n], s, s-1], {n, 100}] (Noe)

{a(n)=if(n<1, 0, numdiv(n)+n%2-1)} /* Michael Somos, Sep 03 2006 */

{a(n)=if(n<1, 0, sumdiv(n,d, d!=2))} /* Michael Somos, Sep 03 2006 */

a(n) = A023645(n) + 1. - T. D. Noe, Jan 18 2006
a(n) = tau(n) - [n is even] = A000005(n) - A059841(n). Proof: gcd(Fib(m), Fib(n)) = Fib(gcd(m, n)) and Fib(2) = 1. - Olivier Wittenberg, following a conjecture of Ralf Stephan, Sep 28 2004
The number of divisors of n excluding 2.
a(2n) = A066660(n). a(2n-1) = A099774(n). - Michael Somos, Sep 03 2006
a(3*2^(Prime(n-1)-1)) = 2n + 1 for n > 3. a(3*2^A068499[n]) = 2n + 1, where A068499(n) = {1,2,3,4,6,10,12,16,18,...}. - Alexander Adamchuk, Sep 15 2006

Corrected and extended by Sascha Kurz, Jan 26 2003

Edited by N. J. A. Sloane, Sep 14 2006. Some of the comments and formulas may need to be adjusted to reflect the new offset.

A069930 Number of integers of the form (n+k)/(n-k) with 1 <= k <= n-1.

Original entry on oeis.org

0, 1, 2, 2, 2, 4, 2, 3, 4, 4, 2, 6, 2, 4, 6, 4, 2, 7, 2, 6, 6, 4, 2, 8, 4, 4, 6, 6, 2, 10, 2, 5, 6, 4, 6, 10, 2, 4, 6, 8, 2, 10, 2, 6, 10, 4, 2, 10, 4, 7, 6, 6, 2, 10, 6, 8, 6, 4, 2, 14, 2, 4, 10, 6, 6, 10, 2, 6, 6, 10, 2, 13, 2, 4, 10, 6, 6, 10, 2, 10, 8, 4, 2, 14, 6, 4, 6, 8, 2, 16, 6, 6, 6, 4, 6

Offset: 1

Benoit Cloitre, May 05 2002

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

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A032741, A069283, A000005 (tau), A001227, A006218, A060831, A066743.

A066660(n) - 1.

Table[Count[Table[(n+k)/(n-k),{k,n-1}],?IntegerQ],{n,100}] (* _Harvey P. Dale, Jun 04 2019 *)

for(n=1,150,print1(sum(i=1,n-1,if((n+i)%(n-i),0,1)),","))

{a(n) = if( n<1, 0, numdiv(2*n) - 2)} /* Michael Somos, Aug 30 2012 */

a(n) = A032741(n) + A069283(n) = A000005(n) - 1 + A001227(n) - 1 = tau(n) + A001227(n) - 2. - Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), Jul 13 2002
Asymptotic formula: since sum(k=1, n, a(k)) = sum(k=1, n, tau(k)) + sum(k=1, n, A001227(k)) - 2*n = A006218(n) + A060831(n) - 2*n = 2*A006218(n) - A006218(floor(n/2)) - 2*n with A006218(0) = 0, A006218(n) = sum(k=1, n, tau(k)) and now, by Dirichlet's asymptotic expression A006218(n) = n*log(n) + n*(2*gamma-1) + O(n^theta) (gamma = 0.57721..; 1/4 <= theta < 1/2), we have sum(k=1, n, a(k)) = 2*n*log(n) - (n/2)*log(n) + o(n*log(n)) = 1.5*n*log(n) + o(n*log(n)) - Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), Jul 13 2002
a(n) = tau(2*n) - 2. - Michael Somos, Aug 30 2012
Sum_{k=1..n} a(k) ~ n/2 * (3*log(n) + log(2) + 6*gamma - 7), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Feb 13 2019

A234306 a(n) = n + 1 - d(2n), where d(n) is the number of divisors of n.

Original entry on oeis.org

0, 0, 0, 1, 2, 1, 4, 4, 4, 5, 8, 5, 10, 9, 8, 11, 14, 10, 16, 13, 14, 17, 20, 15, 20, 21, 20, 21, 26, 19, 28, 26, 26, 29, 28, 25, 34, 33, 32, 31, 38, 31, 40, 37, 34, 41, 44, 37, 44, 42, 44, 45, 50, 43, 48, 47, 50, 53, 56, 45, 58, 57, 52, 57, 58, 55, 64, 61

Offset: 1

Wesley Ivan Hurt, Dec 22 2013

Number of partitions of 2n into exactly two parts: (2n-i,i) such that i does not divide 2n-i. Complement of A066660.

Number of positive integers k <= n, such that k does not divide 2n-k. For example, a(12) = 5 since there are 5 positive integers k less than or equal to 12 that do not divide 2*12-k. They are 5, 7, 9, 10, and 11. - Wesley Ivan Hurt, Jun 24 2021

			a(6) = 1; In this case, 2(6) = 12 has exactly 6 partitions into two parts: (11,1), (10,2), (9,3), (8,4), (7,5), (6,6).  Note that 5 does not divide 7 but the smallest parts of the other partitions divide their corresponding largest parts.  Therefore, a(6) = 1.

Muniru A Asiru, Table of n, a(n) for n = 1..5000
Index entries for sequences related to partitions

Cf. A000005, A066660.

List([1..10^4], n->n+1-Tau(2*n)); # Muniru A Asiru, Feb 04 2018

with(numtheory); A234306:=n->n + 1 - tau(2*n); seq(A234306(n), n=1..100);

Table[n + 1 - DivisorSigma[0, 2n], {n, 100}]

a(n) = n + 1 - numdiv(2*n); \\ Michel Marcus, Dec 23 2013

a(n) = n + 1 - A000005(2n).
a(n) = n - A066660(n).
a(n) = Sum_{i=1..n | i does not divide 2n-i} 1.

A138652 Number of differences (not all necessarily distinct) between consecutive divisors of 2n which are also divisors of 2n.

Original entry on oeis.org

1, 2, 3, 3, 2, 5, 2, 4, 5, 5, 2, 7, 2, 4, 6, 5, 2, 8, 2, 6, 7, 4, 2, 9, 3, 4, 7, 5, 2, 11, 2, 6, 6, 4, 3, 11, 2, 4, 6, 7, 2, 10, 2, 6, 10, 4, 2, 11, 3, 8, 6, 6, 2, 11, 5, 6, 6, 4, 2, 15, 2, 4, 9, 7, 4, 9, 2, 6, 6, 8, 2, 14, 2, 4, 9, 6, 2, 11, 2, 8, 9, 4, 2, 15, 4, 4, 6, 6, 2, 17, 3, 6, 6, 4, 4, 13, 2, 6, 9

Offset: 1

Leroy Quet, May 15 2008

nonn

For n = any odd positive integer, there are no differences (between consecutive divisors of n) that divide n.

			From _Antti Karttunen_, Feb 20 2023: (Start)
Divisors of 2*12 = 24 are: [1, 2, 3, 4, 6, 8, 12, 24]. Their first differences are: [1, 1, 1, 2, 2, 4, 12], all 7 which are divisors of 24, thus a(12) = 7.
Divisors of 2*35 = 70 are: [1, 2, 5, 7, 10, 14, 35, 70]. Their first differences are: 1, 3, 2, 3, 4, 21, 35, of which 1, 2 and 35 are divisors of 70, thus a(35) = 3.
Divisors of 2*65 = 130 are: [1, 2, 5, 10, 13, 26, 65, 130]. Their first differences are: 1, 3, 5, 3, 13, 39, 65, of which 1, 5, 13 and 65 are divisors of 130, thus a(65) = 4.
(End)

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A060741, A060763, A066660, A360118.

A138652 := proc(n) local a,dvs,i ; a := 0 ; dvs := sort(convert(numtheory[divisors](2*n),list)) ; for i from 2 to nops(dvs) do if (2*n) mod ( op(i,dvs)-op(i-1,dvs) ) = 0 then a := a+1 ; fi ; od: a ; end: seq(A138652(n),n=1..120) ; # R. J. Mathar, May 20 2008

a = {}; For[n = 2, n < 200, n = n + 2, b = Table[Divisors[n][[i + 1]] - Divisors[n][[i]], {i, 1, Length[Divisors[n]] - 1}]; AppendTo[a, Length[Select[b, Mod[n, # ] == 0 &]]]]; a (* Stefan Steinerberger, May 18 2008 *)

A138652(n) = { n = 2*n; my(d=divisors(n), erot = vector(#d-1, k, d[k+1] - d[k])); sum(i=1,#erot,!(n%erot[i])); }; \\ Antti Karttunen, Feb 20 2023

a(n) + A360118(2n) = A000005(2n)-1, i.e., a(n) = A066660(n) - A360118(2*n). - Reference to a wrong A-number replaced with A360118 by Antti Karttunen, Feb 20 2023

More terms from Stefan Steinerberger and R. J. Mathar, May 18 2008

Definition edited and clarified by Antti Karttunen, Feb 20 2023

A113241 a(n) = Sum_{k=1..n} (tau(2*k) - 1).

Original entry on oeis.org

0, 1, 3, 6, 9, 12, 17, 20, 24, 29, 34, 37, 44, 47, 52, 59, 64, 67, 75, 78, 85, 92, 97, 100, 109, 114, 119, 126, 133, 136, 147, 150, 156, 163, 168, 175, 186, 189, 194, 201, 210, 213, 224, 227, 234, 245, 250, 253, 264, 269, 277, 284, 291, 294, 305, 312, 321, 328

Offset: 0

Paul Barry, Oct 19 2005

Alois P. Heinz, Table of n, a(n) for n = 0..10000

Partial sums of A066660.

Cf. A000005.

a:= proc(n) option remember; `if`(n<1, 0,
      a(n-1)+numtheory[tau](2*n)-1)
    end:
seq(a(n), n=0..57);  # Alois P. Heinz, Jun 16 2025

CoefficientList[ Series[ Sum[x^k(1 - x^(3k))/((1 - x^k)(1 - x^(2k))), {k, 60}], {x, 0, 57}]/(1 - x), x]

G.f.: (1/(1-x)) * Sum_{k>0} x^k*(1-x^(3*k))/((1-x^k)*(1-x^(2*k))).

a(n) = -n + Sum_{k=1..n} tau(2*k).

More terms from Robert G. Wilson v, Oct 21 2005

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A076984 Number of Fibonacci numbers that are divisors of the n-th Fibonacci number.

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A069930 Number of integers of the form (n+k)/(n-k) with 1 <= k <= n-1.

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A234306 a(n) = n + 1 - d(2n), where d(n) is the number of divisors of n.

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A138652 Number of differences (not all necessarily distinct) between consecutive divisors of 2n which are also divisors of 2n.

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A113241 a(n) = Sum_{k=1..n} (tau(2*k) - 1).

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