cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-8 of 8 results.

A204892 Least k such that n divides s(k)-s(j) for some j in [1,k), where s(k)=prime(k).

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Jan 20 2012

Keywords

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
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

Examples

			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.

Links

Crossrefs

Cf. A000040, A204890.

Programs

  • 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 J. C. Moses, Jan 27 2012 *)
  • 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

A205451 The index j

Original entry on oeis.org

1, 1, 2, 1, 2, 2, 1, 3, 2, 1, 1, 2, 3, 5, 2, 3, 4, 2, 1, 1, 4, 2, 5, 2, 12, 2, 1, 3, 1, 4, 1, 3, 2, 4, 1, 6, 9, 1, 2, 6, 4, 4, 10, 1, 4, 5, 3, 6, 2, 11, 2, 2, 13, 1, 5, 3, 2, 1, 1, 2
Offset: 1

Views

Author

Clark Kimberling, Jan 27 2012

Keywords

Comments

For a guide to related sequences, see A204892.

Crossrefs

Cf. A205450, A204892.

Programs

  • Mathematica
    (See the program at A205450.)

A205454 s(k)-s(j), where (k,j) is the least pair for which n divides s(k)-s(j), and s(j)=Fibonacci(2j).

Original entry on oeis.org

2, 2, 18, 20, 5, 18, 7, 136, 18, 20, 143, 984, 13, 322, 46365, 2576, 34, 18, 6764, 20, 966, 374, 322, 984, 75025, 52, 54, 2576, 986, 317790, 2178308, 832032, 46365, 34, 17710, 46224, 4181, 6764, 2178306, 2440, 123, 966, 39603, 2178308, 317790
Offset: 1

Views

Author

Clark Kimberling, Jan 27 2012

Keywords

Comments

For a guide to related sequences, see A204892.

Crossrefs

Cf. A205450, A204892.

Programs

  • Mathematica
    (See the program at A205450.)

A205448 Ordered differences of even-indexed Fibonacci numbers.

Original entry on oeis.org

2, 7, 5, 20, 18, 13, 54, 52, 47, 34, 143, 141, 136, 123, 89, 376, 374, 369, 356, 322, 233, 986, 984, 979, 966, 932, 843, 610, 2583, 2581, 2576, 2563, 2529, 2440, 2207, 1597, 6764, 6762, 6757, 6744, 6710, 6621, 6388, 5778, 4181, 17710, 17708
Offset: 1

Views

Author

Clark Kimberling, Jan 27 2012

Keywords

Comments

For a guide to related sequences, see A204892.

Examples

			a(1)=s(2)-s(1)=3-1=2
a(2)=s(3)-s(1)=8-1=7
a(3)=s(3)-s(2)=8-3=5
a(4)=s(4)-s(1)=21-1=20
a(5)=s(4)-s(2)=21-3=18

Crossrefs

Cf. A205450, A204892.

Programs

  • Mathematica
    (See the program at A205450.)

A205452 Least s(k) such that n divides s(k)-s(j) for some j

Original entry on oeis.org

3, 3, 21, 21, 8, 21, 8, 144, 21, 21, 144, 987, 21, 377, 46368, 2584, 55, 21, 6765, 21, 987, 377, 377, 987, 121393, 55, 55, 2584, 987, 317811, 2178309, 832040, 46368, 55, 17711, 46368, 6765, 6765, 2178309, 2584, 144, 987, 46368, 2178309, 317811
Offset: 1

Views

Author

Clark Kimberling, Jan 27 2012

Keywords

Comments

For a guide to related sequences, see A204892.

Crossrefs

Cf. A205450, A204892.

Programs

  • Mathematica
    (See the program at A205450.)

A205453 s(A205451), where s(j)=Fibonacci(2j).

Original entry on oeis.org

1, 1, 3, 1, 3, 3, 1, 8, 3, 1, 1, 3, 8, 55, 3, 8, 21, 3, 1, 1, 21, 3, 55, 3, 46368, 3, 1, 8, 1, 21, 1, 8, 3, 21, 1, 144, 2584, 1, 3, 144, 21, 21, 6765, 1, 21, 55, 8, 144, 3, 17711, 3, 3, 121393, 1, 55, 8, 3, 1, 1, 3
Offset: 1

Views

Author

Clark Kimberling, Jan 27 2012

Keywords

Comments

For a guide to related sequences, see A204892.

Crossrefs

Cf. A205450, A204892.

Programs

  • Mathematica
    (See the program at A205450.)

A205455 a(n) = (1/n) * A205454(n).

Original entry on oeis.org

2, 1, 6, 5, 1, 3, 1, 17, 2, 2, 13, 82, 1, 23, 3091, 161, 2, 1, 356, 1, 46, 17, 14, 41, 3001, 2, 2, 92, 34, 10593, 70268, 26001, 1405, 1, 506, 1284, 113, 178, 55854, 61, 3, 23, 921, 49507, 7062, 7, 1, 963, 138, 6002, 2006552, 1, 3706, 1, 122, 46, 1795336
Offset: 1

Views

Author

Clark Kimberling, Jan 27 2012

Keywords

Comments

For a guide to related sequences, see A204892.

Crossrefs

Cf. A205450, A204892.

Programs

  • Mathematica
    (See the program at A205450.)

A205449 Least h such that n divides the h-th difference between distinct even-indexed Fibonacci numbers; the differences are ordered as in A205448.

Original entry on oeis.org

1, 1, 5, 4, 3, 5, 2, 13, 5, 4, 11, 23, 6, 20, 57, 31, 10, 5, 37, 4, 25, 17, 20, 23, 78, 8, 7, 31, 22, 82, 106, 94, 57, 10, 46, 61, 45, 37, 107, 34, 14, 25, 65, 106, 82, 20, 9, 61, 38, 89, 173, 8, 91, 7, 41, 31, 173, 22, 407, 467
Offset: 1

Views

Author

Clark Kimberling, Jan 27 2012

Keywords

Comments

For a guide to related sequences, see A204892.

Crossrefs

Cf. A205450, A204892.

Programs

  • Mathematica
    (See the program at A205450.)
Showing 1-8 of 8 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.