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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). 249
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

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<h'<=n.  Then s(i+h) is congruent mod n to s(i+h'), so that there exist j and k in N such that j<k and n divides s(k)-s(j).  Let k(n) be the least k for which such j exists, and let j(n)=j.  The pair (k,j) will be called the "least pair for which n divides s(k)-s(j)."  (However, starting with "least j for which there is a k" yields pairs (k,j) which differ from those already described.)

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)=p(n)*p(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)=(p(n+1)+p(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

LINKS

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

EXAMPLE

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) s.t. 2 divides s(k)-s(j): (3,2), so a(2)=3.

least (k,j) s.t. 3 divides s(k)-s(j): (3,1), so a(3)=3.

MATHEMATICA

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 Moses, Jan 27 2012 *)

PROG

(PARI) 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

CROSSREFS

Cf. A000040, A204890.

Sequence in context: A029086 A070046 A130120 * A164512 A127434 A205402

Adjacent sequences:  A204889 A204890 A204891 * A204893 A204894 A204895

KEYWORD

nonn

AUTHOR

Clark Kimberling, Jan 20 2012

STATUS

approved

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Last modified October 31 05:50 EDT 2014. Contains 248845 sequences.