A204924 -id:A204924 - cp's OEIS frontend

A204892 Least k such that n divides s(k)-s(j) for some j in [1,k), where s(k)=prime(k).

2, 3, 3, 4, 4, 5, 7, 5, 5, 6, 6, 7, 10, 7, 7, 8, 8, 9, 13, 9, 9, 10, 16, 10, 16, 10, 10, 11, 11, 12, 19, 12, 20, 12, 12, 13, 22, 13, 13, 14, 14, 15, 24, 15, 15, 16, 25, 16, 26, 16, 16, 17, 29, 17, 30, 17, 17, 18, 18, 19, 31, 19, 32, 19, 19, 20, 33, 20, 20, 21

Offset: 1

Clark Kimberling, Jan 20 2012

nonn

Suppose that (s(i)) is a strictly increasing sequence in the set N of positive integers. For i in N, let r(h) be the residue of s(i+h)-s(i) mod n, for h=1,2,...,n+1. There are at most n distinct residues r(h), so that there must exist numbers h and h' such that r(h)=r(h'), where 0<=h

Corollary: for each n, there are infinitely many pairs (j,k) such that n divides s(k)-s(j), and this result holds if s is assumed unbounded, rather than strictly increasing.

Guide to related sequences:

...

s(n)=prime(n), primes

... k(n), j(n): A204892, A204893

... s(k(n)),s(j(n)): A204894, A204895

... s(k(n))-s(j(n)): A204896, A204897

s(n)=prime(n+1), odd primes

... k(n), j(n): A204900, A204901

... s(k(n)),s(j(n)): A204902, A204903

... s(k(n))-s(j(n)): A109043(?), A000034(?)

s(n)=prime(n+2), primes >=5

... k(n), j(n): A204908, A204909

... s(k(n)),s(j(n)): A204910, A204911

... s(k(n))-s(j(n)): A109043(?), A000034(?)

s(n)=prime(n)*prime(n+1) product of consecutive primes

... k(n), j(n): A205146, A205147

... s(k(n)),s(j(n)): A205148, A205149

... s(k(n))-s(j(n)): A205150, A205151

s(n)=(prime(n+1)+prime(n+2))/2: averages of odd primes

... k(n), j(n): A205153, A205154

... s(k(n)),s(j(n)): A205372, A205373

... s(k(n))-s(j(n)): A205374, A205375

s(n)=2^(n-1), powers of 2

... k(n), j(n): A204979, A001511(?)

... s(k(n)),s(j(n)): A204981, A006519(?)

... s(k(n))-s(j(n)): A204983(?), A204984

s(n)=2^n, powers of 2

... k(n), j(n): A204987, A204988

... s(k(n)),s(j(n)): A204989, A140670(?)

... s(k(n))-s(j(n)): A204991, A204992

s(n)=C(n+1,2), triangular numbers

... k(n), j(n): A205002, A205003

... s(k(n)),s(j(n)): A205004, A205005

... s(k(n))-s(j(n)): A205006, A205007

s(n)=n^2, squares

... k(n), j(n): A204905, A204995

... s(k(n)),s(j(n)): A204996, A204997

... s(k(n))-s(j(n)): A204998, A204999

s(n)=(2n-1)^2, odd squares

... k(n), j(n): A205378, A205379

... s(k(n)),s(j(n)): A205380, A205381

... s(k(n))-s(j(n)): A205382, A205383

s(n)=n(3n-1), pentagonal numbers

... k(n), j(n): A205138, A205139

... s(k(n)),s(j(n)): A205140, A205141

... s(k(n))-s(j(n)): A205142, A205143

s(n)=n(2n-1), hexagonal numbers

... k(n), j(n): A205130, A205131

... s(k(n)),s(j(n)): A205132, A205133

... s(k(n))-s(j(n)): A205134, A205135

s(n)=C(2n-2,n-1), central binomial coefficients

... k(n), j(n): A205010, A205011

... s(k(n)),s(j(n)): A205012, A205013

... s(k(n))-s(j(n)): A205014, A205015

s(n)=(1/2)C(2n,n), (1/2)*(central binomial coefficients)

... k(n), j(n): A205386, A205387

... s(k(n)),s(j(n)): A205388, A205389

... s(k(n))-s(j(n)): A205390, A205391

s(n)=n(n+1), oblong numbers

... k(n), j(n): A205018, A205028

... s(k(n)),s(j(n)): A205029, A205030

... s(k(n))-s(j(n)): A205031, A205032

s(n)=n!, factorials

... k(n), j(n): A204932, A204933

... s(k(n)),s(j(n)): A204934, A204935

... s(k(n))-s(j(n)): A204936, A204937

s(n)=n!!, double factorials

... k(n), j(n): A204982, A205100

... s(k(n)),s(j(n)): A205101, A205102

... s(k(n))-s(j(n)): A205103, A205104

s(n)=3^n-2^n

... k(n), j(n): A205000, A205107

... s(k(n)),s(j(n)): A205108, A205109

... s(k(n))-s(j(n)): A205110, A205111

s(n)=Fibonacci(n+1)

... k(n), j(n): A204924, A204925

... s(k(n)),s(j(n)): A204926, A204927

... s(k(n))-s(j(n)): A204928, A204929

s(n)=Fibonacci(2n-1)

... k(n), j(n): A205442, A205443

... s(k(n)),s(j(n)): A205444, A205445

... s(k(n))-s(j(n)): A205446, A205447

s(n)=Fibonacci(2n)

... k(n), j(n): A205450, A205451

... s(k(n)),s(j(n)): A205452, A205453

... s(k(n))-s(j(n)): A205454, A205455

s(n)=Lucas(n)

... k(n), j(n): A205114, A205115

... s(k(n)),s(j(n)): A205116, A205117

... s(k(n))-s(j(n)): A205118, A205119

s(n)=n*(2^(n-1))

... k(n), j(n): A205122, A205123

... s(k(n)),s(j(n)): A205124, A205125

... s(k(n))-s(j(n)): A205126, A205127

s(n)=ceiling[n^2/2]

... k(n), j(n): A205394, A205395

... s(k(n)),s(j(n)): A205396, A205397

... s(k(n))-s(j(n)): A205398, A205399

s(n)=floor[(n+1)^2/2]

... k(n), j(n): A205402, A205403

... s(k(n)),s(j(n)): A205404, A205405

... s(k(n))-s(j(n)): A205406, A205407

			Let s(k)=prime(k).  As in A204890, the ordering of differences s(k)-s(j), follows from the arrangement shown here:
k...........1..2..3..4..5...6...7...8...9
s(k)........2..3..5..7..11..13..17..19..23
...
s(k)-s(1)......1..3..5..9..11..15..17..21..27
s(k)-s(2).........2..4..8..10..14..16..20..26
s(k)-s(3)............2..6..8...12..14..18..24
s(k)-s(4)...............4..6...10..12..16..22
...
least (k,j) such that 1 divides s(k)-s(j) for some j is (2,1), so a(1)=2.
least (k,j) such that 2 divides s(k)-s(j): (3,2), so a(2)=3.
least (k,j) such that 3 divides s(k)-s(j): (3,1), so a(3)=3.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A000040, A204890.

s[n_] := s[n] = Prime[n]; z1 = 400; z2 = 50;
Table[s[n], {n, 1, 30}]          (* A000040 *)
u[m_] := u[m] = Flatten[Table[s[k] - s[j],
   {k, 2, z1}, {j, 1, k - 1}]][[m]]
Table[u[m], {m, 1, z1}]          (* A204890 *)
v[n_, h_] := v[n, h] = If[IntegerQ[u[h]/n], h, 0]
w[n_] := w[n] = Table[v[n, h], {h, 1, z1}]
d[n_] := d[n] = First[Delete[w[n],
   Position[w[n], 0]]]
Table[d[n], {n, 1, z2}]          (* A204891 *)
k[n_] := k[n] = Floor[(3 + Sqrt[8 d[n] - 1])/2]
m[n_] := m[n] = Floor[(-1 + Sqrt[8 n - 7])/2]
j[n_] := j[n] = d[n] - m[d[n]] (m[d[n]] + 1)/2
Table[k[n], {n, 1, z2}]          (* A204892 *)
Table[j[n], {n, 1, z2}]          (* A204893 *)
Table[s[k[n]], {n, 1, z2}]       (* A204894 *)
Table[s[j[n]], {n, 1, z2}]       (* A204895 *)
Table[s[k[n]] - s[j[n]], {n, 1, z2}]     (* A204896 *)
Table[(s[k[n]] - s[j[n]])/n, {n, 1, z2}] (* A204897 *)
(* Program 2: generates A204892 and A204893 rapidly *)
s = Array[Prime[#] &, 120];
lk = Table[NestWhile[# + 1 &, 1, Min[Table[Mod[s[[#]] - s[[j]], z], {j, 1, # - 1}]] =!= 0 &], {z, 1, Length[s]}]
Table[NestWhile[# + 1 &, 1, Mod[s[[lk[[j]]]] - s[[#]], j] =!= 0 &], {j, 1, Length[lk]}]
(* Peter J. C. Moses, Jan 27 2012 *)

a(n)=forprime(p=n+2,,forstep(k=p%n,p-1,n,if(isprime(k), return(primepi(p))))) \\ Charles R Greathouse IV, Mar 20 2013

A204922 Ordered differences of Fibonacci numbers.

Original entry on oeis.org

1, 2, 1, 4, 3, 2, 7, 6, 5, 3, 12, 11, 10, 8, 5, 20, 19, 18, 16, 13, 8, 33, 32, 31, 29, 26, 21, 13, 54, 53, 52, 50, 47, 42, 34, 21, 88, 87, 86, 84, 81, 76, 68, 55, 34, 143, 142, 141, 139, 136, 131, 123, 110, 89, 55, 232, 231, 230, 228, 225, 220, 212, 199, 178

Offset: 1

Clark Kimberling, Jan 21 2012

For a guide to related sequences, see A204892. For numbers not in A204922, see A050939.
From Emanuele Munarini, Mar 29 2012: (Start)
Diagonal elements = Fibonacci numbers F(n+1) (A000045)
First column = Fibonacci numbers - 1 (A000071);
Second column = Fibonacci numbers - 2 (A001911);
Row sums = n*F(n+3) - F(n+2) + 2 (A014286);
Central coefficients = F(2*n+1) - F(n+1) (A096140).
(End)

			a(1) = s(2) - s(1) = F(3) - F(2) = 2-1 = 1, where F=A000045;
a(2) = s(3) - s(1) = F(4) - F(2) = 3-1 = 2;
a(3) = s(3) - s(2) = F(4) - F(3) = 3-2 = 1;
a(4) = s(4) - s(1) = F(5) - F(2) = 5-1 = 4.
From _Emanuele Munarini_, Mar 29 2012: (Start)
Triangle begins:
   1;
   2,  1;
   4,  3,  2;
   7,  6,  5,  3;
  12, 11, 10,  8,  5;
  20, 19, 18, 16, 13,  8;
  33, 32, 31, 29, 26, 21, 13;
  54, 53, 52, 50, 47, 42, 34, 21;
  88, 87, 86, 84, 81, 76, 68, 55, 34;
  ... (End)

G. C. Greubel, Rows n=1..100 of triangle, flattened

Cf. A204924, A204892.

/* As triangle */ [[Fibonacci(n+2)-Fibonacci(k+1) : k in [1..n]]: n in [1.. 15]]; // Vincenzo Librandi, Aug 04 2015

Mathematica
```
(See the program at A204924.)
```

create_list(fib(n+3)-fib(k+2),n,0,20,k,0,n); /* Emanuele Munarini, Mar 29 2012 */

{T(n,k) = fibonacci(n+2) - fibonacci(k+1)};
for(n=1,15, for(k=1,n, print1(T(n,k), ", "))) \\ G. C. Greubel, Feb 03 2019

[[fibonacci(n+2) - fibonacci(k+1) for k in (1..n)] for n in (1..15)] # G. C. Greubel, Feb 03 2019

From Emanuele Munarini, Mar 29 2012: (Start)
T(n,k) = Fibonacci(n+2) - Fibonacci(k+1).
T(n,k) = Sum_{i=k..n} Fibonacci(i+1). (End)

A050939 Numbers that are not the sum of consecutive Fibonacci numbers.

Original entry on oeis.org

9, 14, 15, 17, 22, 23, 24, 25, 27, 28, 30, 35, 36, 37, 38, 39, 40, 41, 43, 44, 45, 46, 48, 49, 51, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 69, 70, 71, 72, 73, 74, 75, 77, 78, 79, 80, 82, 83, 85, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100, 101

Offset: 1

N. J. A. Sloane, Jan 02 2000

nonn

From Clark Kimberling, Dec 16 2009: (Start)
(1) This is the ordered sequence of positive numbers that are not the difference between two Fibonacci numbers; see A007298 for a proof.
(2) Let s=(1,2,1,4,2,1,7,4,2,1,12,7,4,2,1,...) be the lengths of runs of consecutive numbers missing from A050939. Is s=A104582? (End)

Cf. A007298, A204922, A204924.

(See A204924, which generates an ordered list of differences of Fibonacci numbers, as in A204922.)

A204928 s(k(n)) - s(j(n)), where (s(k(n)), s(j(n))) is the least pair of distinct Fibonacci numbers for which n divides s(k(n)) - s(j(n)).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 18, 10, 11, 12, 13, 42, 225, 16, 34, 18, 19, 20, 21, 88, 230, 144, 50, 26, 54, 84, 29, 2550, 31, 32, 33, 34, 17710, 144, 555, 76, 4173, 2440, 123, 42, 86, 88, 225, 230, 47, 144, 343, 50, 2550, 52, 53, 54, 55, 2576, 228, 232, 121304

Offset: 1

Clark Kimberling, Jan 21 2012

nonn

The ordering of the pairs (s(k(n)), s(j(n))) is given by A204922. For a guide to related sequences, see A204892.

G. C. Greubel, Table of n, a(n) for n = 1..1000

Cf. A204900, A204892, A204929.

Mathematica
```
(See the program at A204924.)
```

A204929 (s(k(n)) - s(j(n)))/n, where (s(k(n)), s(j(n))) is the least pair of distinct Fibonacci numbers for which n divides s(k(n)) - s(j(n)).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 3, 15, 1, 2, 1, 1, 1, 1, 4, 10, 6, 2, 1, 2, 3, 1, 85, 1, 1, 1, 1, 506, 4, 15, 2, 107, 61, 3, 1, 2, 2, 5, 5, 1, 3, 7, 1, 50, 1, 1, 1, 1, 46, 4, 4, 2056, 451

Offset: 1

Clark Kimberling, Jan 21 2012

nonn

For a guide to related sequences, see A204892.

G. C. Greubel, Table of n, a(n) for n = 1..1000

Cf. A204924, A204892, A204928.

Mathematica
```
(See the program at A204924.)
```

A204926 Least Fibonacci number f such that n divides f-g for some Fibonacci number g satisfying g < f.

Original entry on oeis.org

2, 3, 5, 5, 8, 8, 8, 13, 21, 13, 13, 13, 21, 55, 233, 21, 55, 21, 21, 21, 34, 89, 233, 233, 55, 34, 55, 89, 34, 2584, 34, 34, 34, 55, 17711, 233, 610, 89, 4181, 2584, 144, 55, 89, 89, 233, 233, 55, 233, 377, 55, 2584, 55, 55, 55, 89, 2584, 233, 233, 121393

Offset: 1

Clark Kimberling, Jan 21 2012

nonn

For a guide to related sequences, see A204892.

G. C. Greubel, Table of n, a(n) for n = 1..1000

Cf. A204924, A204892, A000045.

Mathematica
```
(See the program at A204924.)
```

A204927 The number s(j) such that n divides s(k)-s(j), where s(j) is the (j+1)-st Fibonacci number and k is the least positive integer for which such a j>0 exists.

Original entry on oeis.org

1, 1, 2, 1, 3, 2, 1, 5, 3, 3, 2, 1, 8, 13, 8, 5, 21, 3, 2, 1, 13, 1, 3, 89, 5, 8, 1, 5, 5, 34, 3, 2, 1, 21, 1, 89, 55, 13, 8, 144, 21, 13, 3, 1, 8, 3, 8, 89, 34, 5, 34, 3, 2, 1, 34, 8, 5, 1, 89, 1597

Offset: 1

Clark Kimberling, Jan 21 2012

nonn

For a guide to related sequences, see A204892.

G. C. Greubel, Table of n, a(n) for n = 1..1000

Cf. A204924, A204925, A204892, A000045.

Mathematica
```
(See the program at A204924.)
```

A204923 Least k such that n divides A204922(k), the k-th difference of two distinct Fibonacci numbers.

Original entry on oeis.org

1, 2, 5, 4, 9, 8, 7, 14, 18, 13, 12, 11, 20, 34, 60, 19, 35, 18, 17, 16, 27, 37, 58, 65, 32, 26, 29, 40, 25, 128, 24, 23, 22, 35, 191, 65, 87, 42, 141, 131, 52, 34, 39, 37, 60, 58, 33, 65, 74, 32, 128, 31, 30, 29, 44, 125, 59, 56, 286, 226

Offset: 1

Clark Kimberling, Jan 21 2012

nonn

For a guide to related sequences, see A204892.

G. C. Greubel, Table of n, a(n) for n = 1..1000

Cf. A204922, A204924, A204892.

Mathematica
```
(See the program at A204924.)
```

Showing 1-9 of 9 results.

cp's OEIS Frontend

A204925 a(n) is the index j < k such that n divides s(k) - s(j), where k is the least index (A204924) for which such j exists, and s=(1,2,3,5,8,13,...), the Fibonacci numbers.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A204892 Least k such that n divides s(k)-s(j) for some j in [1,k), where s(k)=prime(k).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

A204922 Ordered differences of Fibonacci numbers.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Mathematica

Maxima

PARI

Sage

Formula

A050939 Numbers that are not the sum of consecutive Fibonacci numbers.

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

A204928 s(k(n)) - s(j(n)), where (s(k(n)), s(j(n))) is the least pair of distinct Fibonacci numbers for which n divides s(k(n)) - s(j(n)).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

A204929 (s(k(n)) - s(j(n)))/n, where (s(k(n)), s(j(n))) is the least pair of distinct Fibonacci numbers for which n divides s(k(n)) - s(j(n)).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

A204926 Least Fibonacci number f such that n divides f-g for some Fibonacci number g satisfying g < f.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

A204927 The number s(j) such that n divides s(k)-s(j), where s(j) is the (j+1)-st Fibonacci number and k is the least positive integer for which such a j>0 exists.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs