A205386 -id:A205386 - 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

A205387 The index j

Original entry on oeis.org

1, 1, 1, 2, 3, 5, 2, 2, 1, 5, 6, 5, 4, 5, 5, 2, 1, 3, 6, 4, 5, 6, 3, 5, 3, 7, 2, 5, 3, 5, 10, 2, 6, 1, 1, 3, 1, 6, 7, 4, 2, 5, 14, 6, 3, 3, 5, 5, 1, 4

Offset: 1

Clark Kimberling, Jan 27 2012

nonn

For a guide to related sequences, see A204892.

Cf. A205386, A204892.

Mathematica
```
(See the program at A205386.)
```

A205388 Least s(k) such that n divides s(k)-s(j) for some j

Original entry on oeis.org

3, 3, 10, 35, 35, 462, 10, 35, 10, 1716, 1716, 462, 126, 462, 1716, 35, 35, 24310, 1716, 6435, 462, 1716, 92378, 462, 35, 24310, 462, 462, 126, 1716, 352716, 35, 1716, 35, 1716, 24310, 24310, 1716, 6435, 6435, 126, 462, 77558760, 24310, 24310

Offset: 1

Clark Kimberling, Jan 27 2012

nonn

For a guide to related sequences, see A204892.

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

Cf. A205386, A204892.

N:= 100: # for a(1)..a(N)
S:=proc(j) option remember; binomial(2*j,j)/2 end proc:
A:= Vector(N): T:= {$1..N}:
for k from 2 while T <> {} do
  for j from 1 to k-1 while T <>{} do
    w:= S(k)-S(j);
    d:= select(t -> w mod t = 0, T);
    A[convert(d,list)]:= S(k);
    T:= T minus d;
  od;
od;
convert(A,list); # Robert Israel, Aug 28 2019

Mathematica
```
(See the program at A205386.)
```

A205390 s(k)-s(j), where (k,j) is the least pair for which n divides s(k)-s(j), and s(j)=(1/2)C(2j,j).

Original entry on oeis.org

2, 2, 9, 32, 25, 336, 7, 32, 9, 1590, 1254, 336, 91, 336, 1590, 32, 34, 24300, 1254, 6400, 336, 1254, 92368, 336, 25, 22594, 459, 336, 116, 1590, 260338, 32, 1254, 34, 1715, 24300, 24309, 1254, 4719, 6400, 123, 336, 57500460, 23848, 24300

Offset: 1

Clark Kimberling, Jan 27 2012

nonn

For a guide to related sequences, see A204892.

Cf. A205386, A204892.

Mathematica
```
(See the program at A205386.)
```

A205384 Ordered differences of numbers s(j)=(1/2)C(2j,j).

Original entry on oeis.org

2, 9, 7, 34, 32, 25, 125, 123, 116, 91, 461, 459, 452, 427, 336, 1715, 1713, 1706, 1681, 1590, 1254, 6434, 6432, 6425, 6400, 6309, 5973, 4719, 24309, 24307, 24300, 24275, 24184, 23848, 22594, 17875, 92377, 92375, 92368, 92343, 92252, 91916

Offset: 1

Clark Kimberling, Jan 27 2012

nonn

For a guide to related sequences, see A204892.

			a(1)=s(2)-s(1)=3-1=2
a(2)=s(3)-s(1)=10-1=9
a(3)=s(3)-s(2)=10-3=7
a(4)=s(4)-s(1)=35-1=34
a(5)=s(4)-s(2)=35-3=32

Cf. A205386, A204892.

Mathematica
```
(See the program at A205386.)
```

A205389 s(A205387), where s(j)=(1/2)C(2j,j).

Original entry on oeis.org

1, 1, 1, 3, 10, 126, 3, 3, 1, 126, 462, 126, 35, 126, 126, 3, 1, 10, 462, 35, 126, 462, 10, 126, 10, 1716, 3, 126, 10, 126, 92378, 3, 462, 1, 1, 10, 1, 462, 1716, 35, 3, 126, 20058300, 462, 10, 10, 126, 126, 1, 35

Offset: 1

Clark Kimberling, Jan 27 2012

nonn

For a guide to related sequences, see A204892.

Cf. A205386, A204892.

Mathematica
```
(See the program at A205386.)
```

A205391 (1/n)*A205390(n).

Original entry on oeis.org

2, 1, 3, 8, 5, 56, 1, 4, 1, 159, 114, 28, 7, 24, 106, 2, 2, 1350, 66, 320, 16, 57, 4016, 14, 1, 869, 17, 12, 4, 53, 8398, 1, 38, 1, 49, 675, 657, 33, 121, 160, 3, 8, 1337220, 542, 540, 2008, 110642, 7, 35, 128

Offset: 1

Clark Kimberling, Jan 27 2012

nonn

For a guide to related sequences, see A204892.

Cf. A205386, A204892.

Mathematica
```
(See the program at A205386.)
```

A205385 Least h such that n divides the h-th difference between distinct numbers (1/2)C(2j,j), as ordered in A205384.

Original entry on oeis.org

1, 1, 2, 5, 6, 15, 3, 5, 2, 20, 21, 15, 10, 15, 20, 5, 4, 31, 21, 25, 15, 21, 39, 15, 6, 35, 12, 15, 9, 20, 55, 5, 21, 4, 16, 31, 29, 21, 28, 25, 8, 15, 105, 34, 31, 39, 71, 15, 16, 25

Offset: 1

Clark Kimberling, Jan 27 2012

nonn

For a guide to related sequences, see A204892.

Cf. A205386, A204892.

Mathematica
```
(See the program at A205386.)
```

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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).

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PARI

A205387 The index j

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A205388 Least s(k) such that n divides s(k)-s(j) for some j

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A205390 s(k)-s(j), where (k,j) is the least pair for which n divides s(k)-s(j), and s(j)=(1/2)C(2j,j).

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A205384 Ordered differences of numbers s(j)=(1/2)C(2j,j).

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A205389 s(A205387), where s(j)=(1/2)C(2j,j).

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A205391 (1/n)*A205390(n).

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A205385 Least h such that n divides the h-th difference between distinct numbers (1/2)C(2j,j), as ordered in A205384.

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