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.

A205100 The index jA204982) for which such j exists.

Original entry on oeis.org

1, 1, 2, 3, 3, 2, 1, 4, 3, 4, 5, 3, 2, 1, 3, 6, 3, 5, 6, 4, 6, 10, 2, 6, 5, 1, 8, 6, 5, 5, 7, 8, 5, 3, 7, 8, 5, 2, 6, 4, 5, 6, 6, 10, 3, 2, 1, 6, 7, 9, 3, 1, 9, 8, 10, 6, 6, 4, 1, 7
Offset: 1

Views

Author

Clark Kimberling, Jan 22 2012

Keywords

Comments

For a guide to related sequences, see A204892.

Crossrefs

Cf. A204982, A204892.

Programs

  • Mathematica
    (See the program at A204982.)

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

A205103 k!!-j!!, where (k!!,j!!) is the least pair of double factorials for which n divides their difference.

Original entry on oeis.org

1, 2, 6, 12, 5, 6, 7, 40, 45, 40, 33, 12, 13, 14, 45, 336, 102, 90, 57, 40, 336, 42240, 46, 336, 3825, 104, 3456, 336, 645105, 90, 279, 3456, 33, 102, 840, 3456, 46065, 3838, 897, 40, 369, 336, 2026977, 42240, 45, 46, 47, 336, 10290, 9450, 102, 104
Offset: 1

Views

Author

Clark Kimberling, Jan 22 2012

Keywords

Comments

For a guide to related sequences, see A204892.

Crossrefs

Cf. A204982, A204892.

Programs

  • Mathematica
    (See the program at A204982.)

A205101 Least k!! such that n divides k!!-j!! for some j

Original entry on oeis.org

2, 3, 8, 15, 8, 8, 8, 48, 48, 48, 48, 15, 15, 15, 48, 384, 105, 105, 105, 48, 384, 46080, 48, 384, 3840, 105, 3840, 384, 645120, 105, 384, 3840, 48, 105, 945, 3840, 46080, 3840, 945, 48, 384, 384, 2027025, 46080, 48, 48, 48, 384, 10395, 10395, 105
Offset: 1

Views

Author

Clark Kimberling, Jan 22 2012

Keywords

Comments

For a guide to related sequences, see A204892.

Crossrefs

Cf. A204982, A204892.

Programs

  • Mathematica
    (See the program at A204982.)

A205102 The number j!! such that n divides k!!-j!!>0, where k is the least positive integer for which such a j exists.

Original entry on oeis.org

1, 1, 2, 3, 3, 2, 1, 8, 3, 8, 15, 3, 2, 1, 3, 48, 3, 15, 48, 8, 48, 3840, 2, 48, 15, 1, 384, 48, 15, 15, 105, 384, 15, 3, 105, 384, 15, 2, 48, 8, 15, 48, 48, 3840, 3, 2, 1, 48, 105, 945, 3, 1, 945, 384, 3840, 48, 48, 8, 1, 105
Offset: 1

Views

Author

Clark Kimberling, Jan 22 2012

Keywords

Comments

For a guide to related sequences, see A204892.

Crossrefs

Cf. A204982, A204892.

Programs

  • Mathematica
    (See the program at A204982.)

A205104 a(n) = (1/n)*A205103(n).

Original entry on oeis.org

1, 1, 2, 3, 1, 1, 1, 5, 5, 4, 3, 1, 1, 1, 3, 21, 6, 5, 3, 2, 16, 1920, 2, 14, 153, 4, 128, 12, 22245, 3, 9, 108, 1, 3, 24, 96, 1245, 101, 23, 1, 9, 8, 47139, 960, 1, 1, 1, 7, 210, 189, 2, 2, 650160, 64, 768, 6, 1, 177964, 16, 14
Offset: 1

Views

Author

Clark Kimberling, Jan 22 2012

Keywords

Comments

For a guide to related sequences, see A204892.

Crossrefs

Cf. A204982, A204892.

Programs

  • Mathematica
    (* See the program at A204982 *)

A204912 Ordered differences of double factorials.

Original entry on oeis.org

1, 2, 1, 7, 6, 5, 14, 13, 12, 7, 47, 46, 45, 40, 33, 104, 103, 102, 97, 90, 57, 383, 382, 381, 376, 369, 336, 279, 944, 943, 942, 937, 930, 897, 840, 561, 3839, 3838, 3837, 3832, 3825, 3792, 3735, 3456, 2895, 10394, 10393, 10392, 10387, 10380
Offset: 1

Views

Author

Clark Kimberling, Jan 22 2012

Keywords

Comments

For a guide to related sequences, see A204892.
As a triangle with rows {1}, {2,1}, {7,6,5}, ..., evidently the first diagonal is A007911 = {1,1,5,7,33,57,279,...} (up to an offset). - L. Edson Jeffery, Jan 24 2012

Examples

			a(1)=2!!-1!!=2-1=1
a(2)=3!!-1!!=3-1=2
a(3)=3!!-2!!=3-2=1
a(4)=4!!-1!!=8-1=7
a(5)=4!!-2!!=8-2=6

Crossrefs

Cf. A204982, A204892.
Cf. A007911.

Programs

  • Mathematica
    (See the program at A204982.)

A204913 Least k such that n divides the k-th difference between distinct double factorials.

Original entry on oeis.org

1, 2, 5, 9, 6, 5, 4, 14, 13, 14, 15, 9, 8, 7, 13, 27, 18, 20, 21, 14, 27, 65, 12, 27, 41, 16, 44, 27, 83, 20, 28, 44, 15, 18, 35, 44, 60, 38, 34, 14, 26, 27, 97, 65, 13, 12, 11, 27, 52, 54, 18, 16, 129, 44, 65, 27, 21, 109, 29, 35
Offset: 1

Views

Author

Clark Kimberling, Jan 22 2012

Keywords

Comments

For a guide to related sequences, see A204892.

Crossrefs

Cf. A204982, A204892.

Programs

  • Mathematica
    (See the program at A204982.)
Showing 1-8 of 8 results.

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