A205000 -id:A205000 - 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

A205110 s(k)-s(j), where (s(k),s(j)) is the least pair of numbers given by s(j)=3^j-2^j which n divides their difference.

Original entry on oeis.org

4, 4, 18, 4, 60, 18, 14, 64, 18, 60, 660, 60, 2054, 14, 60, 64, 646, 18, 646, 60, 210, 660, 46, 192, 600, 2054, 19170, 1848, 1586126, 60, 17112, 64, 660, 646, 210, 6300, 19166, 646, 6240, 600, 1394, 210, 508174, 660, 6300, 46, 5640, 192, 2058, 600

Offset: 1

Clark Kimberling, Jan 22 2012

nonn

For a guide to related sequences, see A204892.

Cf. A205000, A204892.

Mathematica
```
(See the program at A205000.)
```

A205108 Least s(k) such that n divides s(k)-s(j) for some j

Original entry on oeis.org

5, 5, 19, 5, 65, 19, 19, 65, 19, 65, 665, 65, 2059, 19, 65, 65, 665, 19, 665, 65, 211, 665, 65, 211, 665, 2059, 19171, 2059, 1586131, 65, 19171, 65, 665, 665, 211, 6305, 19171, 665, 6305, 665, 2059, 211, 527345, 665, 6305, 65, 6305, 211, 2059, 665

Offset: 1

Clark Kimberling, Jan 22 2012

nonn

For a guide to related sequences, see A204892.

Cf. A205000, A204892.

Mathematica
```
(See the program at A205000.)
```

A205109 The number s(j) such that n divides s(k)-s(j)>0, where k is the least positive integer for which such a j exists, and s(j)=3^j-2^j.

Original entry on oeis.org

1, 1, 1, 1, 5, 1, 5, 1, 1, 5, 5, 5, 5, 5, 5, 1, 19, 1, 19, 5, 1, 5, 19, 19, 65, 5, 1, 211, 5, 5, 2059, 1, 5, 19, 1, 5, 5, 19, 65, 65, 665, 1, 19171, 5, 5, 19, 665, 19, 1, 65, 19, 65, 175099, 1, 5, 211, 19, 5, 211, 5

Offset: 1

Clark Kimberling, Jan 22 2012

nonn

For a guide to related sequences, see A204892.

Cf. A205000, A204892.

Mathematica
```
(See the program at A205000.)
```

A205111 a(n) = A205110(n)/n.

Original entry on oeis.org

4, 2, 6, 1, 12, 3, 2, 8, 2, 6, 60, 5, 158, 1, 4, 4, 38, 1, 34, 3, 10, 30, 2, 8, 24, 79, 710, 66, 54694, 2, 552, 2, 20, 19, 6, 175, 518, 17, 160, 15, 34, 5, 11818, 15, 140, 1, 120, 4, 42, 12, 40, 120, 807662, 355, 12, 33, 336, 27347, 26880, 1

Offset: 1

Clark Kimberling, Jan 22 2012

nonn

For a guide to related sequences, see A204892.

Cf. A204892, A205000, A205110.

Mathematica
```
(See the program at A205000.)
```

A205105 Ordered differences of numbers 3^j-2^j, as in A001047.

Original entry on oeis.org

4, 18, 14, 64, 60, 46, 210, 206, 192, 146, 664, 660, 646, 600, 454, 2058, 2054, 2040, 1994, 1848, 1394, 6304, 6300, 6286, 6240, 6094, 5640, 4246, 19170, 19166, 19152, 19106, 18960, 18506, 17112, 12866, 58024, 58020, 58006, 57960, 57814

Offset: 1

Clark Kimberling, Jan 22 2012

nonn

For a guide to related sequences, see A204892.

			a(1)=s(2)-s(1)=(3^2-2^2)-(3^1-2^1)=4
a(2)=s(3)-s(1)=19-1=18
a(3)=s(3)-s(2)=19-5=14
a(4)=s(4)-s(1)=65-1=64

Cf. A205000, A204892.

Mathematica
```
(See the program at A205000.)
```

A205106 Least k such that n divides the k-th difference between distinct pairs of numbers 3^j-2^j.

Original entry on oeis.org

1, 1, 2, 1, 5, 2, 3, 4, 2, 5, 12, 5, 17, 3, 5, 4, 13, 2, 13, 5, 7, 12, 6, 9, 14, 17, 29, 20, 68, 5, 35, 4, 12, 13, 7, 23, 30, 13, 25, 14, 21, 7, 64, 12, 23, 6, 27, 9, 16, 14, 18, 25, 116, 29, 12, 20, 31, 68, 71, 5

Offset: 1

Clark Kimberling, Jan 22 2012

nonn

For a guide to related sequences, see A204892.

Cf. A205000, A204892.

Mathematica
```
(See the program at A205000.)
```

Showing 1-8 of 8 results.

cp's OEIS Frontend

A205107 The index jA205000) for which such j exists, and s(k)=3^k-2^k.

Views

Author

Keywords

Comments

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

A205110 s(k)-s(j), where (s(k),s(j)) is the least pair of numbers given by s(j)=3^j-2^j which n divides their difference.

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

A205108 Least s(k) such that n divides s(k)-s(j) for some j

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

A205109 The number s(j) such that n divides s(k)-s(j)>0, where k is the least positive integer for which such a j exists, and s(j)=3^j-2^j.

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

A205111 a(n) = A205110(n)/n.

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

A205105 Ordered differences of numbers 3^j-2^j, as in A001047.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A205106 Least k such that n divides the k-th difference between distinct pairs of numbers 3^j-2^j.

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica