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.

A204988 The index j < k such that n divides 2^k - 2^j, where k is the least index (A204987) for which such j exists.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 1, 2, 1, 1, 1, 4, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 1, 2, 1, 1, 1, 5, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 1, 2, 1, 1, 1, 4, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 1, 2, 1, 1, 1, 6, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 1, 2, 1, 1, 1, 4, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 1, 2, 1, 1, 1, 5, 1, 1, 1, 2, 1, 1, 1, 3, 1
Offset: 1

Views

Author

Clark Kimberling, Jan 21 2012

Keywords

Comments

For a guide to related sequences, see A204892.

Examples

			(See example at A204987.)

Links

Crossrefs

Cf. A007814, A204892, A204987.

Programs

Formula

a(n) = A007814(n) + (1-(-1)^n)/2 (conjecture). - Velin Yanev, Nov 14 2016.
From Andrew Howroyd, Aug 08 2018: (Start)
The above conjecture is true because the definition of this sequence and A204987 requires j to be at least 1 and 2^k - 2^j can be written 2^j*(2^(k-j) - 1).
a(n) = max(1, A007814(n)). (End)
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 3/2. - Amiram Eldar, Oct 22 2022

Extensions

More terms from Antti Karttunen, Nov 19 2017
Keyword:mult added by Andrew Howroyd, Aug 08 2018

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

A204991 2^k-2^j, where (2^k,2^j) is the least pair of distinct positive powers of 2 for which n divides 2^k-2^j.

Original entry on oeis.org

2, 2, 6, 4, 30, 6, 14, 8, 126, 30, 2046, 12, 8190, 14, 30, 16, 510, 126, 524286, 60, 126, 2046, 4094, 24, 2097150, 8190, 524286, 28, 536870910, 30, 62, 32, 2046, 510, 8190, 252, 137438953470, 524286, 8190, 120, 2097150, 126, 32766, 4092, 8190
Offset: 1

Views

Author

Clark Kimberling, Jan 21 2012

Keywords

Comments

For a guide to related sequences, see A204892.

Crossrefs

Cf. A204987, A204892, A204990=(1/2)(A204991).

Programs

  • Mathematica
    (See the program at A204987.)

A204985 Ordered differences of numbers 2^k for k>=1.

Original entry on oeis.org

2, 6, 4, 14, 12, 8, 30, 28, 24, 16, 62, 60, 56, 48, 32, 126, 124, 120, 112, 96, 64, 254, 252, 248, 240, 224, 192, 128, 510, 508, 504, 496, 480, 448, 384, 256, 1022, 1020, 1016, 1008, 992, 960, 896, 768, 512, 2046, 2044, 2040, 2032, 2016, 1984, 1920
Offset: 1

Views

Author

Clark Kimberling, Jan 21 2012

Keywords

Comments

A204985=2*A130328. For a guide to related sequences, see A204892.

Crossrefs

Cf. A204987, A204892.

Programs

  • Mathematica
    (See the program at A204987.)

A204986 Least k such that n divides A204985(k), the k-th difference between numbers of the form 2^k.

Original entry on oeis.org

1, 1, 2, 3, 7, 2, 4, 6, 16, 7, 46, 5, 67, 4, 7, 10, 29, 16, 154, 12, 16, 46, 56, 9, 191, 67, 154, 8, 379, 7, 11, 15, 46, 29, 67, 23, 631, 154, 67, 18, 191, 16, 92, 57, 67, 56, 254, 14, 211, 191
Offset: 1

Views

Author

Clark Kimberling, Jan 21 2012

Keywords

Comments

A204986(2n-1)=A204939(2n-1) for n>=1. For a guide to related sequences, see A204892.

Crossrefs

Cf. A204987, A204892.

Programs

  • Mathematica
    (See the program at A204987.)

A204989 Least 2^k such that n divides 2^k-2^j for some j

Original entry on oeis.org

4, 4, 8, 8, 32, 8, 16, 16, 128, 32, 2048, 16, 8192, 16, 32, 32, 512, 128, 524288, 64, 128, 2048, 4096, 32, 2097152, 8192, 524288, 32, 536870912, 32, 64, 64, 2048, 512, 8192, 256, 137438953472, 524288, 8192, 128, 2097152, 128, 32768, 4096
Offset: 1

Views

Author

Clark Kimberling, Jan 21 2012

Keywords

Comments

For a guide to related sequences, see A204892.

Crossrefs

Cf. A204892, A204987.

Programs

  • Mathematica
    (See the program at A204987.)

A204990 (1/2)*(A204991).

Original entry on oeis.org

1, 1, 3, 2, 15, 3, 7, 4, 63, 15, 1023, 6, 4095, 7, 15, 8, 255, 63, 262143, 30, 63, 1023, 2047, 12, 1048575, 4095, 262143, 14, 268435455, 15, 31, 16, 1023, 255, 4095, 126, 68719476735, 262143, 4095, 60, 1048575, 63, 16383, 2046, 4095, 2047
Offset: 1

Views

Author

Clark Kimberling, Jan 21 2012

Keywords

Comments

For a guide to related sequences, see A204892.

Crossrefs

Cf. A204987, A204892, A204991.

Programs

  • Mathematica
    (See the program at A204987.)

A204992 (1/n)*A204991(n).

Original entry on oeis.org

2, 1, 2, 1, 6, 1, 2, 1, 14, 3, 186, 1, 630, 1, 2, 1, 30, 7, 27594, 3, 6, 93, 178, 1, 83886, 315, 19418, 1, 18512790, 1, 2, 1, 62, 15, 234, 7, 3714566310, 13797, 210, 3, 51150, 3, 762, 93, 182, 89, 356962, 1, 85598, 41943
Offset: 1

Views

Author

Clark Kimberling, Jan 21 2012

Keywords

Comments

For a guide to related sequences, see A204892.

Crossrefs

Cf. A204987, A204991.

Programs

  • Mathematica
    (See the program at A204987.)
Showing 1-8 of 8 results.

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