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.

A205028 The index jA205018) for which such j exists, and s(k)=k*(k+1).

Original entry on oeis.org

1, 1, 2, 1, 1, 2, 2, 3, 1, 1, 4, 2, 5, 2, 3, 7, 7, 1, 8, 1, 5, 4, 10, 2, 2, 5, 1, 1, 13, 3, 14, 15, 2, 7, 1, 2, 17, 8, 3, 1, 19, 5, 20, 3, 4, 10, 22, 6, 3, 2, 5, 4, 25, 1, 8, 4, 6, 13, 28, 3
Offset: 1

Views

Author

Clark Kimberling, Jan 22 2012

Keywords

Comments

For a guide to related sequences, see A204892.

Crossrefs

Cf. A205018, A204892.

Programs

  • Mathematica
    (See the program at A205018.)

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

A205031 s(k)-s(j), where (s(k),s(j)) is the least pair of oblong numbers for which n divides their difference.

Original entry on oeis.org

4, 4, 6, 4, 10, 6, 14, 8, 18, 10, 22, 24, 26, 14, 30, 16, 34, 18, 38, 40, 42, 22, 46, 24, 50, 26, 54, 28, 58, 30, 62, 32, 66, 34, 70, 36, 74, 38, 78, 40, 82, 42, 86, 44, 90, 46, 94, 48, 98, 50, 102, 52, 106, 54, 110, 112, 114, 58, 118, 60
Offset: 1

Views

Author

Clark Kimberling, Jan 22 2012

Keywords

Comments

For a guide to related sequences, see A204892.

Crossrefs

Cf. A205018, A204892.

Programs

  • Mathematica
    (See the program at A205018.)

A205032 a(n) = (s(k)-s(j))/n, where (s(k),s(j)) is the least pair of oblong numbers (A002378) for which n divides their difference; a(n) = (1/n)*A205031(n).

Original entry on oeis.org

4, 2, 2, 1, 2, 1, 2, 1, 2, 1, 2, 2, 2, 1, 2, 1, 2, 1, 2, 2, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 2, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 2, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2
Offset: 1

Views

Author

Clark Kimberling, Jan 22 2012

Keywords

Comments

For a guide to related sequences, see A204892.
Even n such that a(n) = 2 are: 2, 12, 20, 56, 72, 132, 156, 240, 272, 380, 552, 812, 992, 1056, 1332, 1640, 1892, 2256, 2756, 3540, 3660, 4032, 4160, 4556, 5112, 5256, 6320, 6972, 7656, 7832, ... - Antti Karttunen, Nov 06 2018

Links

Crossrefs

Cf. A002378, A205018, A205031, A204892.
Cf. also A204897, A204999, A205007.

Programs

  • Mathematica
    (See the program at A205018.)
  • PARI
    A205032(n) = for(k=2,oo,my(sk=k*(k+1)); for(j=1,k-1,if(!((sk-((j+1)*j))%n),return((sk-((j+1)*j))/n)))); \\ Antti Karttunen, Nov 06 2018
  • PARI
    A205032(n) = for(k=sqrtint(n)-1,oo,my(sk=k*(k+1), d); for(j=1,k-1,d=(sk-((j+1)*j)); if(0==(d%n),return(d/n),if(dAntti Karttunen, Nov 06 2018

Extensions

Definition edited and more terms from Antti Karttunen, Nov 06 2018

A205029 Least s(k) such that n divides s(k)-s(j) for some j

Original entry on oeis.org

6, 6, 12, 6, 12, 12, 20, 20, 20, 12, 42, 30, 56, 20, 42, 72, 90, 20, 110, 42, 72, 42, 156, 30, 56, 56, 56, 30, 240, 42, 272, 272, 72, 90, 72, 42, 380, 110, 90, 42, 462, 72, 506, 56, 110, 156, 600, 90, 110, 56, 132, 72, 756, 56, 182, 132, 156, 240, 930, 72
Offset: 1

Views

Author

Clark Kimberling, Jan 22 2012

Keywords

Comments

For a guide to related sequences, see A204892.

Crossrefs

Cf. A205018, A204892.

Programs

  • Mathematica
    (See the program at A205018.)

A205030 The number s(j)=j*(j+1) such that n divides s(k)-s(j)>0, where k is the least positive integer for which such a j exists.

Original entry on oeis.org

2, 2, 6, 2, 2, 6, 6, 12, 2, 2, 20, 6, 30, 6, 12, 56, 56, 2, 72, 2, 30, 20, 110, 6, 6, 30, 2, 2, 182, 12, 210, 240, 6, 56, 2, 6, 306, 72, 12, 2, 380, 30, 420, 12, 20, 110, 506, 42, 12, 6, 30, 20, 650, 2, 72, 20, 42, 182, 812, 12
Offset: 1

Views

Author

Clark Kimberling, Jan 22 2012

Keywords

Comments

For a guide to related sequences, see A204892.

Crossrefs

Cf. A205018, A204892.

Programs

  • Mathematica
    (See the program at A205018.)

A205016 Ordered differences of oblong numbers.

Original entry on oeis.org

4, 10, 6, 18, 14, 8, 28, 24, 18, 10, 40, 36, 30, 22, 12, 54, 50, 44, 36, 26, 14, 70, 66, 60, 52, 42, 30, 16, 88, 84, 78, 70, 60, 48, 34, 18, 108, 104, 98, 90, 80, 68, 54, 38, 20, 130, 126, 120, 112, 102, 90, 76, 60, 42, 22, 154, 150, 144, 136, 126, 114, 100
Offset: 1

Views

Author

Clark Kimberling, Jan 22 2012

Keywords

Comments

For a guide to related sequences, see A204892.

Examples

			a(1)=s(2)-s(1)=6-2=4
a(2)=s(3)-s(1)=12-2=10
a(3)=s(3)-s(2)=12-6=6
a(4)=s(4)-s(1)=
a(5)=s(4)-s(2)=

Crossrefs

Cf. A205018, A204892.

Programs

  • Mathematica
    (See the program at A205018.)

A205017 Least k such that n divides the k-th difference between distinct oblong numbers.

Original entry on oeis.org

1, 1, 3, 1, 2, 3, 5, 6, 4, 2, 14, 8, 20, 5, 13, 28, 35, 4, 44, 11, 26, 14, 65, 8, 17, 20, 16, 7, 104, 13, 119, 120, 23, 35, 22, 12, 170, 44, 31, 11, 209, 26, 230, 18, 40, 65, 275, 34, 39, 17, 50, 25, 350, 16, 74, 49, 61, 104, 434, 24
Offset: 1

Views

Author

Clark Kimberling, Jan 22 2012

Keywords

Comments

For a guide to related sequences, see A204892.

Crossrefs

Cf. A205018, A204892.

Programs

  • Mathematica
    (See the program at A205018.)
Showing 1-8 of 8 results.

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