A205378 -id:A205378 - 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

A205383 a(n) = (1/n)*A205382(n).

Original entry on oeis.org

8, 4, 8, 2, 8, 4, 8, 1, 8, 4, 8, 2, 8, 4, 8, 1, 8, 4, 8, 2, 8, 4, 8, 1, 8, 4, 8, 2, 8, 4, 8, 1, 8, 4, 8, 2, 8, 4, 8, 1, 8, 4, 8, 2, 8, 4, 8, 1, 8, 4, 8, 2, 8, 4, 8, 1, 8, 4, 8, 2

Offset: 1

Clark Kimberling, Jan 26 2012

nonn

Possibly a(n) = (1/n)*lcm(8,n); see A205382.

For a guide to related sequences, see A204892.

Cf. A205378, A204892.

Mathematica
```
(See the program at A205378.)
```

A205379 The index j

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 3, 1, 2, 2, 5, 1, 6, 3, 1, 2, 8, 2, 9, 2, 1, 5, 11, 1, 3, 6, 2, 3, 14, 1, 15, 4, 3, 8, 2, 2, 18, 9, 4, 2, 20, 1, 21, 5, 1, 11, 23, 1, 4, 3, 6, 6, 26, 2, 1, 3, 7, 14, 29, 1

Offset: 1

Clark Kimberling, Jan 26 2012

nonn

For a guide to related sequences, see A204892.

Cf. A205378, A204892.

Mathematica
```
(See the program at A205378.)
```

A205382 s(k)-s(j), where (s(k),s(j)) is the least such pair for which n divides their difference, and s(j)=(2j-1)^2.

Original entry on oeis.org

8, 8, 24, 8, 40, 24, 56, 8, 72, 40, 88, 24, 104, 56, 120, 16, 136, 72, 152, 40, 168, 88, 184, 24, 200, 104, 216, 56, 232, 120, 248, 32, 264, 136, 280, 72, 296, 152, 312, 40, 328, 168, 344, 88, 360, 184, 376, 48, 392, 200, 408, 104, 424, 216, 440, 56, 456

Offset: 1

Clark Kimberling, Jan 26 2012

nonn

Duplicate of A109049? For a guide to related sequences, see A204892.

Cf. A205383, A205378, A204892, A109049.

Mathematica
```
(See the program at A205378.)
```

a(n) = n*A205383(n). - Luce ETIENNE, Feb 19 2020

A205376 Ordered differences of distinct odd squares, stored in triangle.

Original entry on oeis.org

8, 24, 16, 48, 40, 24, 80, 72, 56, 32, 120, 112, 96, 72, 40, 168, 160, 144, 120, 88, 48, 224, 216, 200, 176, 144, 104, 56, 288, 280, 264, 240, 208, 168, 120, 64, 360, 352, 336, 312, 280, 240, 192, 136, 72, 440, 432, 416, 392, 360, 320, 272, 216, 152

Offset: 1

Clark Kimberling, Jan 26 2012

a(n) = 8*A049777(n). For a guide to related sequences, see A204892.

Triangle T(n,k), k>=0, n>=1, read by rows, given by T(n,k) = (2*n+1)^2 - (2*k+1)^2. - Philippe Deléham, Mar 07 2013

			a(1) = s(2)-s(1) =  9-1 = 8,
a(2) = s(3)-s(1) = 25-1 = 24,
a(3) = s(3)-s(2) = 25-9 = 16,
a(4) = s(4)-s(1) = 49-1 = 48,
a(5) = s(4)-s(2) = 49-9 = 40.
Triangle begins:
8
24,  16
48,  40,  24
80,  72,  56,  32
120, 112, 96,  72,  40
168, 160, 144, 120, 88, 48, ... - _Philippe Deléham_, Mar 07 2013

Cf. A205378, A204892.

Mathematica
```
(See the program at A205378.)
```

A205380 Least s(k) such that n divides s(k)-s(j) for some j

Original entry on oeis.org

9, 9, 25, 9, 49, 25, 81, 9, 81, 49, 169, 25, 225, 81, 121, 25, 361, 81, 441, 49, 169, 169, 625, 25, 225, 225, 225, 81, 961, 121, 1089, 81, 289, 361, 289, 81, 1521, 441, 361, 49, 1849, 169, 2025, 169, 361, 625, 2401, 49, 441, 225, 529, 225, 3025, 225

Offset: 1

Clark Kimberling, Jan 26 2012

nonn

For a guide to related sequences, see A204892.

Cf. A205378, A204892.

Mathematica
```
(See the program at A205378.)
```

A205381 s(A205379), where s(j)=(2j-1)^2.

Original entry on oeis.org

1, 1, 1, 1, 9, 1, 25, 1, 9, 9, 81, 1, 121, 25, 1, 9, 225, 9, 289, 9, 1, 81, 441, 1, 25, 121, 9, 25, 729, 1, 841, 49, 25, 225, 9, 9, 1225, 289, 49, 9, 1521, 1, 1681, 81, 1, 441, 2025, 1, 49, 25, 121, 121, 2601, 9, 1, 25, 169, 729, 3249, 1

Offset: 1

Clark Kimberling, Jan 26 2012

nonn

For a guide to related sequences, see A204892.

Cf. A205378, A204892.

Mathematica
```
(See the program at A205378.)
```

A205377 Least h such that n divides the h-th difference between distinct odd primes, as ordered in A205376.

Original entry on oeis.org

1, 1, 2, 1, 5, 2, 9, 1, 8, 5, 20, 2, 27, 9, 11, 3, 44, 8, 54, 5, 16, 20, 77, 2, 24, 27, 23, 9, 119, 11, 135, 10, 31, 44, 30, 8, 189, 54, 40, 5, 230, 16, 252, 20, 37, 77, 299, 4, 49, 24, 61, 27, 377, 23, 46, 9, 73, 119, 464, 11

Offset: 1

Clark Kimberling, Jan 26 2012

nonn

For a guide to related sequences, see A204892.

Cf. A205378, A204892.

Mathematica
```
(See the program at A205378.)
```

Showing 1-8 of 8 results.

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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A205383 a(n) = (1/n)*A205382(n).

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A205379 The index j

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A205382 s(k)-s(j), where (s(k),s(j)) is the least such pair for which n divides their difference, and s(j)=(2j-1)^2.

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Formula

A205376 Ordered differences of distinct odd squares, stored in triangle.

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A205380 Least s(k) such that n divides s(k)-s(j) for some j

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A205381 s(A205379), where s(j)=(2j-1)^2.

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A205377 Least h such that n divides the h-th difference between distinct odd primes, as ordered in A205376.

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