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.

A205131 The index jA205130) for which such j exists, and s(k)=2*k^2-k, the k-th hexagonal number.

Original entry on oeis.org

1, 1, 2, 1, 1, 3, 1, 1, 2, 3, 2, 2, 3, 1, 3, 1, 4, 1, 4, 2, 5, 2, 5, 3, 6, 9, 1, 7, 7, 3, 7, 1, 8, 12, 3, 5, 9, 4, 2, 5, 10, 6, 10, 1, 1, 5, 11, 2, 2, 4, 3, 8, 13, 6, 6, 5, 2, 21, 14, 2
Offset: 1

Views

Author

Clark Kimberling, Jan 25 2012

Keywords

Comments

For a guide to related sequences, see A204892.

Crossrefs

Cf. A205130, A204892.

Programs

  • Mathematica
    (See the program at A205130.)

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

A205134 s(k)-s(j), where (s(k),s(j)) is the least such pair for which n divides their difference, and s(j)=2*j^2-j, the j-th hexagonal number.

Original entry on oeis.org

5, 14, 9, 44, 5, 30, 14, 152, 9, 30, 22, 60, 13, 14, 30, 560, 17, 90, 38, 60, 21, 22, 46, 216, 25, 78, 27, 140, 29, 30, 62, 2144, 33, 102, 105, 108, 37, 38, 39, 280, 41, 210, 86, 44, 90, 46, 94, 624, 147, 350, 51, 156, 53, 54, 165, 280, 114, 174, 118, 60
Offset: 1

Views

Author

Clark Kimberling, Jan 25 2012

Keywords

Comments

For a guide to related sequences, see A204892.

Crossrefs

Cf. A205130, A204892.

Programs

  • Mathematica
    (See the program at A205130.)

A205128 Ordered differences of distinct hexagonal numbers.

Original entry on oeis.org

5, 14, 9, 27, 22, 13, 44, 39, 30, 17, 65, 60, 51, 38, 21, 90, 85, 76, 63, 46, 25, 119, 114, 105, 92, 75, 54, 29, 152, 147, 138, 125, 108, 87, 62, 33, 189, 184, 175, 162, 145, 124, 99, 70, 37, 230, 225, 216, 203, 186, 165, 140, 111, 78, 41, 275, 270, 261
Offset: 1

Views

Author

Clark Kimberling, Jan 25 2012

Keywords

Comments

For a guide to related sequences, see A204892.

Examples

			a(1)=s(2)-s(1)=6-1=5
a(2)=s(3)-s(1)=15-1=14
a(3)=s(3)-s(2)=15-6=9
a(4)=s(4)-s(1)=28-1=27
a(5)=s(4)-s(2)=28-6=22

Crossrefs

Cf. A205130, A204892.

Programs

  • Mathematica
    (See the program at A205130.)

A205132 Least s(k) such that n divides s(k)-s(j) for some j

Original entry on oeis.org

6, 15, 15, 45, 6, 45, 15, 153, 15, 45, 28, 66, 28, 15, 45, 561, 45, 91, 66, 66, 66, 28, 91, 231, 91, 231, 28, 231, 120, 45, 153, 2145, 153, 378, 120, 153, 190, 66, 45, 325, 231, 276, 276, 45, 91, 91, 325, 630, 153, 378, 66, 276, 378, 120, 231, 325, 120
Offset: 1

Views

Author

Clark Kimberling, Jan 25 2012

Keywords

Comments

For a guide to related sequences, see A204892.

Crossrefs

Cf. A205130, A204892.

Programs

  • Mathematica
    (See the program at A205130.)

A205133 The number s(j) such that n divides s(k)-s(j), where s(j)=j*(2*j-1) and k is the least positive integer for which such a j exists.

Original entry on oeis.org

1, 1, 6, 1, 1, 15, 1, 1, 6, 15, 6, 6, 15, 1, 15, 1, 28, 1, 28, 6, 45, 6, 45, 15, 66, 153, 1, 91, 91, 15, 91, 1, 120, 276, 15, 45, 153, 28, 6, 45, 190, 66, 190, 1, 1, 45, 231, 6, 6, 28, 15, 120, 325, 66, 66, 45, 6, 861, 378, 6
Offset: 1

Views

Author

Clark Kimberling, Jan 25 2012

Keywords

Comments

For a guide to related sequences, see A204892.

Crossrefs

Cf. A205130, A204892.

Programs

  • Mathematica
    (See the program at A205130.)

A205135 (1/n)*A205134(n).

Original entry on oeis.org

5, 7, 3, 11, 1, 5, 2, 19, 1, 3, 2, 5, 1, 1, 2, 35, 1, 5, 2, 3, 1, 1, 2, 9, 1, 3, 1, 5, 1, 1, 2, 67, 1, 3, 3, 3, 1, 1, 1, 7, 1, 5, 2, 1, 2, 1, 2, 13, 3, 7, 1, 3, 1, 1, 3, 5, 2, 3, 2, 1
Offset: 1

Views

Author

Clark Kimberling, Jan 25 2012

Keywords

Comments

For a guide to related sequences, see A204892.

Crossrefs

Cf. A205130, A204892.

Programs

  • Mathematica
    (See the program at A205130.)

A205129 Least k such that n divides a difference between distinct hexagonal numbers, ordered as in A205128.

Original entry on oeis.org

1, 2, 3, 7, 1, 9, 2, 29, 3, 9, 5, 12, 6, 2, 9, 121, 10, 16, 14, 12, 15, 5, 20, 48, 21, 54, 4, 52, 28, 9, 35, 497, 36, 90, 24, 33, 45, 14, 8, 71, 55, 61, 65, 7, 16, 20, 77, 138, 30, 82, 13, 63, 91, 27, 51, 71, 23, 252, 119, 12
Offset: 1

Views

Author

Clark Kimberling, Jan 25 2012

Keywords

Comments

For a guide to related sequences, see A204892.

Crossrefs

Cf. A205130, A204892.

Programs

  • Mathematica
    (See the program at A205130.)
Showing 1-8 of 8 results.

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