A204988 -id:A204988 - cp's OEIS frontend

A204892 Least k such that n divides s(k)-s(j) for some j in [1,k), where s(k)=prime(k).

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

Clark Kimberling, Jan 20 2012

nonn

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

			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.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A000040, A204890.

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 *)

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

A204987 Least k such that n divides 2^k - 2^j for some j satisfying 1 <= j < k.

Original entry on oeis.org

2, 2, 3, 3, 5, 3, 4, 4, 7, 5, 11, 4, 13, 4, 5, 5, 9, 7, 19, 6, 7, 11, 12, 5, 21, 13, 19, 5, 29, 5, 6, 6, 11, 9, 13, 8, 37, 19, 13, 7, 21, 7, 15, 12, 13, 12, 24, 6, 22, 21, 9, 14, 53, 19, 21, 6, 19, 29, 59, 6, 61, 6, 7, 7, 13, 11, 67, 10, 23, 13, 36, 9, 10, 37, 21, 20, 31, 13, 40, 8, 55, 21, 83, 8, 9, 15, 29, 13

Offset: 1

Clark Kimberling, Jan 21 2012

nonn

See A204892 for a discussion and guide to related sequences.

			1 divides 2^2 - 2^1, so a(1)=2;
2 divides 2^2 - 2^1, so a(2)=2;
3 divides 2^3 - 2^1, so a(3)=3;
4 divides 2^3 - 2^2, so a(4)=3;
5 divides 2^5 - 2^1, so a(5)=5.

Antti Karttunen, Table of n, a(n) for n = 1..6556

Cf. A000079, A007733, A007814, A054703, A204892, A204988.

s[n_] := s[n] = 2^n; z1 = 1000; z2 = 50;
Table[s[n], {n, 1, 30}]     (* A000079 *)
u[m_] := u[m] = Flatten[Table[s[k] - s[j], {k, 2, z1}, {j, 1, k - 1}]][[m]]
Table[u[m], {m, 1, z1}]     (* A204985 *)
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}]     (* A204986 *)
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}]     (* A204987 *)
Table[j[n], {n, 1, z2}]     (* A204988 *)
Table[s[k[n]], {n, 1, z2}]  (* A204989 *)
Table[s[j[n]], {n, 1, z2}]  (* A140670 ? *)
Table[s[k[n]] - s[j[n]], {n, 1, z2}]     (* A204991 *)
Table[(s[k[n]] - s[j[n]])/n, {n, 1, z2}] (* A204992 *)
%%/2  (* A204990=(1/2)*A204991 *)

A204987etA204988(n) = { my(k=2); while(1,for(j=1,k-1,if(!(((2^k)-(2^j))%n),return([k,j]))); k++); }; \\ (Computes also A204988 at the same time) - Antti Karttunen, Nov 19 2017

a(n)={my(k=valuation(n,2)); max(k, 1) + znorder(Mod(2, n>>k))} \\ Andrew Howroyd, Aug 08 2018

a(n) = max(1, A007814(n)) + A007733(n). - Andrew Howroyd, Aug 08 2018

More terms from Antti Karttunen, Nov 19 2017

A378992 a(n) = A011371(n) - A048881(n); The exponent of the highest power of 2 dividing the n-th factorial minus the exponent of the highest power of 2 dividing n-th Catalan number.

Original entry on oeis.org

0, 0, 0, 1, 2, 2, 2, 4, 6, 6, 6, 7, 8, 8, 8, 11, 14, 14, 14, 15, 16, 16, 16, 18, 20, 20, 20, 21, 22, 22, 22, 26, 30, 30, 30, 31, 32, 32, 32, 34, 36, 36, 36, 37, 38, 38, 38, 41, 44, 44, 44, 45, 46, 46, 46, 48, 50, 50, 50, 51, 52, 52, 52, 57, 62, 62, 62, 63, 64, 64, 64, 66, 68, 68, 68, 69, 70, 70, 70, 73, 76, 76, 76

Offset: 0

Antti Karttunen, Dec 16 2024

Apparently, after the initial three 0's, only terms of A092054 occur, every other as a single copy, and every other in a batch of 3 duplicated terms.

Antti Karttunen, Table of n, a(n) for n = 0..20000
Index entries for sequences related to binary expansion of n

Partial sums of A050605.
Cf. A000108, A000120, A000142, A007814, A011371, A048881.
Cf. also A092054, A204988, A050603, A136480.

A378992[n_] := n - DigitCount[n, 2, 1] - DigitCount[n + 1, 2, 1] + 1;
Array[A378992, 100, 0] (* or *)
MapIndexed[#2[[1]] - # &, Total[Partition[DigitCount[Range[0, 100], 2, 1], 2, 1], {2}]] (* Paolo Xausa, Dec 28 2024 *)

A378992(n) = (1+(n-hammingweight(n)-hammingweight(1+n)));

a(n) = A007814(A000142(n)) - A007814(A000108(n)) = A011371(n) - A048881(n).

a(0) = 0; for n > 0, a(n) = A050605(n-1) + a(n-1), where A050605(n) = A007814(n+1)+A007814(n+2)-1.

Showing 1-3 of 3 results.

cp's OEIS Frontend

A204892 Least k such that n divides s(k)-s(j) for some j in [1,k), where s(k)=prime(k).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

A204987 Least k such that n divides 2^k - 2^j for some j satisfying 1 <= j < k.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

Extensions

A378992 a(n) = A011371(n) - A048881(n); The exponent of the highest power of 2 dividing the n-th factorial minus the exponent of the highest power of 2 dividing n-th Catalan number.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula