A205153 -id:A205153 - 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

A205372 Least s(k) such that n divides s(k)-s(j) for some jA024675.

Original entry on oeis.org

6, 6, 9, 12, 9, 12, 18, 12, 15, 26, 15, 18, 30, 18, 21, 34, 21, 30, 34, 26, 30, 26, 50, 30, 34, 30, 39, 34, 50, 34, 76, 50, 39, 60, 39, 42, 76, 42, 45, 86, 45, 60, 64, 50, 60, 50, 56, 60, 64, 56, 60, 56, 129, 60, 64, 60, 69, 64, 93, 64, 76, 134, 69, 76, 69, 72, 76

Offset: 1

Clark Kimberling, Jan 26 2012

nonn

For a guide to related sequences, see A204892.

a(n) >= n+4, with equality if and only if n+4 is in A024675.

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

Cf. A024675, A205153, A204892.

N:= 200: # for terms before the first > the greatest prime <= N
P:= select(isprime, [seq(i,i=3..N,2)]):
S:= (P[1..-2]+P[2..-1])/2:
f:= proc(n) local T,R,i;
   T:= S mod n;
   R:= {}:
   for i from 1 to nops(T)-1 do
     R:= R union {T[i]};
     if member(T[i+1],R) then return S[i+1] fi;
   od;
   FAIL
end proc:
Res:= NULL:
for n from 1 do
  v:= f(n);
  if v = FAIL then break fi;
  Res:= Res, v
od:
Res; # Robert Israel, Sep 09 2020

Mathematica
```
(See the program at A205153.)
```

A205374 s(k)-s(j), where (s(k),s(j)) is the least such pair for which n divides their difference, and s(j)=(prime(j+1) + prime(j+2))/2.

Original entry on oeis.org

2, 2, 3, 8, 5, 6, 14, 8, 9, 20, 11, 12, 26, 14, 15, 16, 17, 18, 19, 20, 21, 22, 46, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 80, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69

Offset: 1

Clark Kimberling, Jan 26 2012

nonn

For a guide to related sequences, see A204892.

Cf. A205153, A204892.

Mathematica
```
(See the program at A205153.)
```

A204980 Ordered differences of distinct averages of two consecutive odd primes.

Original entry on oeis.org

2, 5, 3, 8, 6, 3, 11, 9, 6, 3, 14, 12, 9, 6, 3, 17, 15, 12, 9, 6, 3, 22, 20, 17, 14, 11, 8, 5, 26, 24, 21, 18, 15, 12, 9, 4, 30, 28, 25, 22, 19, 16, 13, 8, 4, 35, 33, 30, 27, 24, 21, 18, 13, 9, 5, 38, 36, 33, 30, 27, 24, 21, 16, 12, 8, 3, 41, 39, 36, 33, 30, 27, 24, 19

Offset: 1

Clark Kimberling, Jan 26 2012

nonn

For a guide to related sequences, see A204892.

			a(1)=s(2)-s(1)=6-4=2
a(2)=s(3)-s(1)=9-4=5
a(3)=s(3)-s(2)=9-6=3
a(4)=s(4)-s(1)=12-4=8
a(5)=s(4)-s(2)=12-6=6

Cf. A205153, A204892.

Mathematica
```
(See the program at A205153.)
```

A205154 The index j

Original entry on oeis.org

1, 1, 2, 1, 1, 2, 1, 1, 2, 2, 1, 2, 1, 1, 2, 6, 1, 4, 5, 2, 3, 1, 1, 2, 3, 1, 4, 2, 7, 1, 13, 6, 2, 8, 1, 2, 11, 1, 2, 2, 1, 6, 7, 2, 5, 1, 3, 4, 5, 2, 3, 1, 20, 2, 3, 1, 4, 2, 10, 1, 5, 19, 2, 4, 1, 2, 3, 1, 4, 2, 5, 1, 8, 4, 2, 8, 1, 5, 8, 2

Offset: 1

Clark Kimberling, Jan 26 2012

nonn

For a guide to related sequences, see A204892.

Cf. A205153, A204892.

Mathematica
```
(See the program at A205153.)
```

A205373 s(A204154), where s(j)=(prime(j+1) + prime(j+2))/2.

Original entry on oeis.org

4, 4, 6, 4, 4, 6, 4, 4, 6, 6, 4, 6, 4, 4, 6, 18, 4, 12, 15, 6, 9, 4, 4, 6, 9, 4, 12, 6, 21, 4, 45, 18, 6, 26, 4, 6, 39, 4, 6, 6, 4, 18, 21, 6, 15, 4, 9, 12, 15, 6, 9, 4, 76, 6, 9, 4, 12, 6, 34, 4, 15, 72, 6, 12, 4, 6, 9, 4, 12, 6, 15, 4, 26, 12, 6, 26, 4, 15, 26, 6

Offset: 1

Clark Kimberling, Jan 26 2012

nonn

For a guide to related sequences, see A204892.

Cf. A205153, A204892.

Mathematica
```
(See the program at A205153.)
```

A205375 (1/n)*A205374(n).

Original entry on oeis.org

2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 1

Clark Kimberling, Jan 26 2012

nonn

For a guide to related sequences, see A204892.

Cf. A205153, A204892.

Mathematica
```
(See the program at A205153.)
```

A205152 Least k such that n divides a difference between distinct averages of consecutive odd primes, as ordered in A204980.

Original entry on oeis.org

1, 1, 3, 4, 2, 5, 11, 4, 8, 23, 7, 12, 29, 11, 17, 42, 16, 32, 41, 23, 31, 22, 79, 30, 39, 29, 49, 38, 85, 37, 184, 84, 47, 113, 46, 57, 182, 56, 68, 212, 67, 111, 127, 80, 110, 79, 94, 109, 125, 93, 108, 92, 426, 107, 123, 106, 140, 122, 241, 121, 176, 454

Offset: 1

Clark Kimberling, Jan 26 2012

nonn

For a guide to related sequences, see A204892.

Cf. A205153, A204892.

Mathematica
```
(See the program at A205153.)
```

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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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A205372 Least s(k) such that n divides s(k)-s(j) for some jA024675.

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A205374 s(k)-s(j), where (s(k),s(j)) is the least such pair for which n divides their difference, and s(j)=(prime(j+1) + prime(j+2))/2.

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A204980 Ordered differences of distinct averages of two consecutive odd primes.

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A205154 The index j

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A205373 s(A204154), where s(j)=(prime(j+1) + prime(j+2))/2.

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A205375 (1/n)*A205374(n).

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A205152 Least k such that n divides a difference between distinct averages of consecutive odd primes, as ordered in A204980.

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