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.

A205139 The index jA205138) for which such j exists, and s(k)=k*(3k-1)/2, the k-th pentagonal number.

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Jan 25 2012

Keywords

Comments

For a guide to related sequences, see A204892.

Crossrefs

Cf. A205138, A204892.

Programs

  • Mathematica
    (See the program at A205138.)

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

A205142 s(k)-s(j), where (s(k),s(j)) is the least such pair for which n divides their difference, and s(j)=j(3j-1)/2, the j-th pentagonal number.

Original entry on oeis.org

4, 4, 21, 4, 10, 30, 7, 16, 144, 10, 11, 48, 13, 70, 30, 16, 17, 144, 19, 80, 21, 22, 23, 48, 50, 130, 1161, 112, 29, 30, 31, 64, 66, 34, 35, 144, 37, 190, 39, 80, 41, 84, 43, 308, 225, 46, 47, 48, 196, 50, 102, 260, 53, 1242, 110, 112, 57, 58, 59, 120
Offset: 1

Views

Author

Clark Kimberling, Jan 25 2012

Keywords

Comments

For a guide to related sequences, see A204892.

Crossrefs

Cf. A205138, A204892.

Programs

  • Mathematica
    (See the program at A205138.)

A205143 (1/n)*A205142(n).

Original entry on oeis.org

4, 2, 7, 1, 2, 5, 1, 2, 16, 1, 1, 4, 1, 5, 2, 1, 1, 8, 1, 4, 1, 1, 1, 2, 2, 5, 43, 4, 1, 1, 1, 2, 2, 1, 1, 4, 1, 5, 1, 2, 1, 2, 1, 7, 5, 1, 1, 1, 4, 1, 2, 5, 1, 23, 2, 2, 1, 1, 1, 2, 1, 5, 4, 1, 1, 1, 1, 7, 1, 1, 1, 2, 1, 5, 1, 5, 2, 5, 1, 1
Offset: 1

Views

Author

Clark Kimberling, Jan 25 2012

Keywords

Comments

For a guide to related sequences, see A204892.

Crossrefs

Cf. A205138, A204892.

Programs

  • Mathematica
    (See the program at A205138.)

A205136 Ordered differences of distinct pentagonal numbers.

Original entry on oeis.org

4, 11, 7, 21, 17, 10, 34, 30, 23, 13, 50, 46, 39, 29, 16, 69, 65, 58, 48, 35, 19, 91, 87, 80, 70, 57, 41, 22, 116, 112, 105, 95, 82, 66, 47, 25, 144, 140, 133, 123, 110, 94, 75, 53, 28, 175, 171, 164, 154, 141, 125, 106, 84, 59, 31, 209, 205, 198, 188, 175
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)=5-1=4
a(2)=s(3)-s(1)=12-1=11
a(3)=s(3)-s(2)=12-5=7
a(4)=s(4)-s(1)=22-1=21
a(5)=s(4)-s(2)=22-5=17

Crossrefs

Cf. A205138, A204892.

Programs

  • Mathematica
    (See the program at A205138.)

A205140 Least s(k) such that n divides s(k)-s(j) for some j

Original entry on oeis.org

5, 5, 22, 5, 22, 35, 12, 51, 145, 22, 12, 70, 35, 92, 35, 51, 22, 145, 70, 92, 22, 92, 35, 70, 51, 247, 1162, 117, 51, 35, 176, 715, 117, 35, 70, 145, 247, 477, 51, 92, 92, 176, 330, 330, 247, 51, 117, 70, 247, 51, 247, 330, 145, 1247, 145, 117, 92, 70
Offset: 1

Views

Author

Clark Kimberling, Jan 25 2012

Keywords

Comments

For a guide to related sequences, see A204892.

Crossrefs

Cf. A204892, A205138.

Programs

  • Mathematica
    (See the program at A205138.)

A205141 The number s(j) such that n divides s(k)-s(j), where s(j)=j*(3*j-1) and k is the least positive integer for which such a j exists.

Original entry on oeis.org

1, 1, 1, 1, 12, 5, 5, 35, 1, 12, 1, 22, 22, 22, 5, 35, 5, 1, 51, 12, 1, 70, 12, 22, 1, 117, 1, 5, 22, 5, 145, 651, 51, 1, 35, 1, 210, 287, 12, 12, 51, 92, 287, 22, 22, 5, 70, 22, 51, 1, 145, 70, 92, 5, 35, 5, 35, 12, 117, 210
Offset: 1

Views

Author

Clark Kimberling, Jan 25 2012

Keywords

Comments

For a guide to related sequences, see A204892.

Crossrefs

Cf. A205138, A204892.

Programs

  • Mathematica
    (See the program at A205138.)

A205137 Least k such that n divides a difference between distinct pentagonal numbers when the differences are ordered as in A205136.

Original entry on oeis.org

1, 1, 4, 1, 6, 8, 3, 15, 37, 6, 2, 19, 10, 25, 8, 15, 5, 37, 21, 24, 4, 28, 9, 19, 11, 75, 352, 30, 14, 8, 55, 231, 34, 7, 20, 37, 78, 150, 13, 24, 27, 53, 105, 95, 70, 12, 35, 19, 72, 11, 76, 98, 44, 380, 41, 30, 26, 18, 54, 103
Offset: 1

Views

Author

Clark Kimberling, Jan 25 2012

Keywords

Comments

For a guide to related sequences, see A204892.

Crossrefs

Cf. A205138, A204892.

Programs

  • Mathematica
    (See the program at A205138.)
Showing 1-8 of 8 results.

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