A205114 -id:A205114 - 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

A205117 The number s(j) such that n divides s(k)-s(j), where s(j) is the j-th Lucas number and k is the least positive integer for which such a j with 0

Original entry on oeis.org

1, 1, 1, 3, 1, 1, 4, 3, 11, 1, 7, 11, 3, 4, 3, 11, 1, 11, 47, 7, 18, 7, 1, 4, 4, 3, 47, 1, 18, 3, 843, 7, 1, 29, 18, 11, 3, 47, 4, 7, 76, 3, 4, 3, 7, 1, 29, 7, 3, 7, 11, 1, 4, 47, 47, 11, 322, 18, 76, 3

Offset: 1

Clark Kimberling, Jan 22 2012

For a guide to related sequences, see A204892.

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

Cf. A205114, A204892.

lucas:= gfun:-rectoproc({a(n)=a(n-1)+a(n-2),a(0)=2, a(1)=1},a(n),remember):
f:= proc(n) local j,k,S,t;
    S:= [];
    for k from 1 do
      t:= lucas(k) mod n;
      if member(t,S,j) then return lucas(j) fi;
      S:= [op(S),t];
    od
end proc:
map(f, [$1..100]); # Robert Israel, Jan 21 2018

Mathematica
```
(See the program at A205114.)
```

Name corrected by Robert Israel, Jan 21 2018

A205118 s(k)-s(j), where (s(k),s(j)) is the least pair of Lucas numbers for which n divides their difference.

Original entry on oeis.org

2, 2, 3, 4, 10, 6, 7, 8, 18, 10, 11, 36, 26, 14, 15, 112, 17, 18, 76, 40, 105, 22, 46, 72, 25, 26, 2160, 28, 29, 120, 1364, 192, 198, 170, 105, 36, 518, 76, 195, 40, 123, 840, 43, 44, 315, 46, 47, 192, 196, 2200, 510, 520, 318, 2160, 275, 112, 3249, 58

Offset: 1

Clark Kimberling, Jan 22 2012

nonn

For a guide to related sequences, see A204892.

Cf. A205114, A204892.

Mathematica
```
(See the program at A205114.)
```

A205112 Ordered differences of Lucas numbers.

Original entry on oeis.org

2, 3, 1, 6, 4, 3, 10, 8, 7, 4, 17, 15, 14, 11, 7, 28, 26, 25, 22, 18, 11, 46, 44, 43, 40, 36, 29, 18, 75, 73, 72, 69, 65, 58, 47, 29, 122, 120, 119, 116, 112, 105, 94, 76, 47, 198, 196, 195, 192, 188, 181, 170, 152, 123, 76, 321, 319, 318, 315, 311, 304, 293

Offset: 1

Clark Kimberling, Jan 22 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)=4-1=3
a(3)=s(3)-s(2)=4-3=1
a(4)=s(4)-s(1)=7-1=6
a(5)=s(4)-s(2)=7-3=4

Cf. A205114, A204892.

Mathematica
```
(See the program at A205114.)
```

A205113 Least k such that n divides the k-th difference between distinct Lucas numbers.

Original entry on oeis.org

1, 1, 2, 5, 7, 4, 9, 8, 20, 7, 14, 26, 17, 13, 12, 41, 11, 20, 44, 25, 42, 19, 22, 31, 18, 17, 113, 16, 27, 38, 119, 49, 46, 52, 42, 26, 68, 44, 48, 25, 54, 80, 24, 23, 59, 22, 35, 49, 47, 109, 71, 67, 58, 113, 63, 41, 132, 34, 87, 38

Offset: 1

Clark Kimberling, Jan 22 2012

The pairs of Lucas numbers are ordered as at A205112. For a guide to related sequences, see A204892.

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

Cf. A205114, A204892.

N:= 100: # to get terms before the first term > N*(N-1)/2
L:= proc(n) option remember; combinat:-fibonacci(n+1)+combinat:-fibonacci(n-1); end proc:
A205112:= [seq(seq(L(j)-L(i),i=1..j-1),j=2..N)]:
M:= N*(N-1)/2:
f:= proc(n) local k;
  for k from 1 to M do if A205112[k] mod n = 0 then return k fi od;
  -1
end proc:
R:= NULL:
for n from 1 do
v:= f(n);
if v = -1 then break fi;
R:= R,v
od:
R; # Robert Israel, Feb 25 2024

Mathematica
```
(See the program at A205114.)
```

A205116 Least s(k) such that n divides s(k)-s(j) for some j

Original entry on oeis.org

3, 3, 4, 7, 11, 7, 11, 11, 29, 11, 18, 47, 29, 18, 18, 123, 18, 29, 123, 47, 123, 29, 47, 76, 29, 29, 2207, 29, 47, 123, 2207, 199, 199, 199, 123, 47, 521, 123, 199, 47, 199, 843, 47, 47, 322, 47, 76, 199, 199, 2207, 521, 521, 322, 2207, 322, 123, 3571

Offset: 1

Clark Kimberling, Jan 22 2012

nonn

For a guide to related sequences, see A204892.

Cf. A205114, A204892.

Mathematica
```
(See the program at A205114.)
```

A205119 (1/n)*A205118(n).

Original entry on oeis.org

2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 3, 2, 1, 1, 7, 1, 1, 4, 2, 5, 1, 2, 3, 1, 1, 80, 1, 1, 4, 44, 6, 6, 5, 3, 1, 14, 2, 5, 1, 3, 20, 1, 1, 7, 1, 1, 4, 4, 44, 10, 10, 6, 40, 5, 2, 57, 1, 13, 2

Offset: 1

Clark Kimberling, Jan 22 2012

nonn

For a guide to related sequences, see A204892.

Cf. A205114, A204892.

Mathematica
```
(See the program at A205114.)
```

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cp's OEIS Frontend

A205115 The index jA205114) for which such j exists, and s=(1,3,4,7,11,18...), the Lucas numbers.

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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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A205117 The number s(j) such that n divides s(k)-s(j), where s(j) is the j-th Lucas number and k is the least positive integer for which such a j with 0

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A205118 s(k)-s(j), where (s(k),s(j)) is the least pair of Lucas numbers for which n divides their difference.

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A205112 Ordered differences of Lucas numbers.

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A205113 Least k such that n divides the k-th difference between distinct Lucas numbers.

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A205116 Least s(k) such that n divides s(k)-s(j) for some j

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A205119 (1/n)*A205118(n).

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