A205010 -id:A205010 - 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

A205014 s(k)-s(j), where (s(k),s(j)) is the least pair of central binomial coefficients for which n divides their difference.

Original entry on oeis.org

1, 4, 18, 4, 5, 18, 14, 64, 18, 50, 2508, 672, 182, 14, 3180, 64, 68, 18, 19, 3180, 672, 2508, 69, 672, 50, 182, 918, 672, 232, 3180, 520676, 64, 2508, 68, 3430, 48600, 48618, 2508, 9438, 12800, 246, 672, 115000920, 2508, 48600, 184736, 3431

Offset: 1

Clark Kimberling, Jan 22 2012

nonn

For a guide to related sequences, see A204892.

Cf. A205010, A204892.

Mathematica
```
(See the program at A205010.)
```

A205008 Ordered differences of central binomial coefficients.

Original entry on oeis.org

1, 5, 4, 19, 18, 14, 69, 68, 64, 50, 251, 250, 246, 232, 182, 923, 922, 918, 904, 854, 672, 3431, 3430, 3426, 3412, 3362, 3180, 2508, 12869, 12868, 12864, 12850, 12800, 12618, 11946, 9438, 48619, 48618, 48614, 48600, 48550, 48368, 47696

Offset: 1

Clark Kimberling, Jan 22 2012

nonn

For a guide to related sequences, see A204892.

			a(1)=s(2)-s(1)=2-1=1
a(2)=s(3)-s(1)=6-1=5
a(3)=s(3)-s(2)=6-2=4
a(4)=s(4)-s(1)=20-1=19
a(5)=s(4)-s(2)=20-2=18

Cf. A205010, A204892.

Mathematica
```
(See the program at A205010.)
```

A205012 Least s(k) such that n divides s(k)-s(j) for some j

Original entry on oeis.org

2, 6, 20, 6, 6, 20, 20, 70, 20, 70, 3432, 924, 252, 20, 3432, 70, 70, 20, 20, 3432, 924, 3432, 70, 924, 70, 252, 924, 924, 252, 3432, 705432, 70, 3432, 70, 3432, 48620, 48620, 3432, 12870, 12870, 252, 924, 155117520, 3432, 48620, 184756, 3432

Offset: 1

Clark Kimberling, Jan 22 2012

nonn

For a guide to related sequences, see A204892.

Cf. A205010, A204892.

Mathematica
```
(See the program at A205010.)
```

A205013 The number s(j)=C(2j-2,j-1) such that n divides s(k)-s(j)>0, where k is the least positive integer for which such a j exists.

Original entry on oeis.org

1, 2, 2, 2, 1, 2, 6, 6, 2, 20, 924, 252, 70, 6, 252, 6, 2, 2, 1, 252, 252, 924, 1, 252, 20, 70, 6, 252, 20, 252, 184756, 6, 924, 2, 2, 20, 2, 924, 3432, 70, 6, 252, 40116600, 924, 20, 20, 1, 252, 2, 20, 6, 3432, 252, 6, 12870, 252, 924, 20, 2333606220, 252

Offset: 1

Clark Kimberling, Jan 22 2012

nonn

For a guide to related sequences, see A204892.

Cf. A205010, A204892.

Mathematica
```
(See the program at A205010.)
```

A205015 (1/n)*A205014(n).

Original entry on oeis.org

1, 2, 6, 1, 1, 3, 2, 8, 2, 5, 228, 56, 14, 1, 212, 4, 4, 1, 1, 159, 32, 114, 3, 28, 2, 7, 34, 24, 8, 106, 16796, 2, 76, 2, 98, 1350, 1314, 66, 242, 320, 6, 16, 2674440, 57, 1080, 4016, 73, 14, 70, 1, 18, 869, 60, 17, 650, 12, 44, 4, 559519620, 53

Offset: 1

Clark Kimberling, Jan 22 2012

nonn

For a guide to related sequences, see A204892.

Cf. A205010, A204892.

Mathematica
```
(See the program at A205010.)
```

A205009 Least k such that n divides the k-th difference between distinct central binomials coefficients.

Original entry on oeis.org

1, 3, 5, 3, 2, 5, 6, 9, 5, 10, 28, 21, 15, 6, 27, 9, 8, 5, 4, 27, 21, 28, 7, 21, 10, 15, 18, 21, 14, 27, 66, 9, 28, 8, 23, 40, 38, 28, 36, 33, 13, 21, 120, 28, 40, 49, 22, 21, 23, 10, 18, 44, 27, 18, 45, 21, 28, 14, 189, 27

Offset: 1

Clark Kimberling, Jan 22 2012

nonn

The ordering of the differences between distinct central binomials coefficients is given by A205008. For a guide to related sequences, see A204892.

Cf. A205010, A204892.

Mathematica
```
(See the program at A205010.)
```

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A205011 The index jA205010) for which such j exists.

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

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A205014 s(k)-s(j), where (s(k),s(j)) is the least pair of central binomial coefficients for which n divides their difference.

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A205008 Ordered differences of central binomial coefficients.

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A205012 Least s(k) such that n divides s(k)-s(j) for some j

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A205013 The number s(j)=C(2j-2,j-1) such that n divides s(k)-s(j)>0, where k is the least positive integer for which such a j exists.

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A205015 (1/n)*A205014(n).

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A205009 Least k such that n divides the k-th difference between distinct central binomials coefficients.

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