A205402 -id:A205402 - 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

A205406 a(n) = s(k)-s(j), where (k,j) is the least pair for which n divides s(k)-s(j), and s(j) = floor((j+1)^2/2)/2.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 18, 10, 11, 12, 13, 14, 15, 16, 34, 18, 19, 20, 21, 22, 23, 24, 50, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 98, 50, 51, 52, 106, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 201, 68

Offset: 1

Clark Kimberling, Jan 27 2012

For a guide to related sequences, see A204892.

			The least k such that 9 divides s(k)-s(j) for some k is k=8, for which j=2: s(8)-s(2) = 20-2 = 18, so a(9)=18.

Antti Karttunen, Table of n, a(n) for n = 1..12004

Cf. A002620, A198293, A205402, A204892.

(See the program at A205402.)
s[m_]:=s[m]=Floor[(m+1)^2/2]/2
A205406[n_]:=(k=2; found=False; While[!found, Do[If[Mod[d=s[k]-s[j], n]==0, found=True; Break[]], {j, k-1}]; k++]; d)
nterms=100; Table[A205406[n], {n, nterms}] (* Paolo Xausa, Dec 03 2021 *)

A002620(n) = ((n^2)>>2);
A002620shiftedleft(n) = A002620(1+n);
A205406(n) = { my(d); for(k=2,oo, for(j=1,k-1,if(!((d=A002620shiftedleft(k)-A002620shiftedleft(j))%n),return(d)))); }; \\ Antti Karttunen, Dec 05 2021

a(n) = n * A198293(n). - Antti Karttunen, Dec 05 2021

Definition corrected by Clark Kimberling, Dec 05 2021

A198293 a(n) = (1/n)*A205406(n).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 3

Offset: 1

Clark Kimberling, Jan 27 2012

nonn

For a guide to related sequences, see A204892.

Paolo Xausa, Table of n, a(n) for n = 1..20000

Cf. A002620, A205402, A205406, A204892.

(See the program at A205402.)
s[m_]:=s[m]=Floor[(m+1)^2/2]/2
A205406[n_]:=(k=2;found=False;While[!found,Do[If[Mod[d=s[k]-s[j],n]==0,found=True;Break[]],{j,k-1}];k++];d)
nterms=100;Table[A205406[n]/n,{n,nterms}] (* Paolo Xausa, Dec 03 2021 *)

A002620(n) = ((n^2)>>2);
A002620shiftedleft(n) = A002620(1+n);
A205406(n) = { my(d); for(k=2,oo, for(j=1,k-1,if(!((d=A002620shiftedleft(k)-A002620shiftedleft(j))%n),return(d)))); };
A198293(n) = (A205406(n)/n); \\ Antti Karttunen, Dec 03 2021

Data section extended up to a(111) by Antti Karttunen, Dec 03 2021

A205403 a(n) is the index j < k such that n divides s(k)-s(j) for some j, where s(j) = floor((j+1)^2/2)/2, and k is the least index for which such a j exists.

Original entry on oeis.org

1, 2, 1, 2, 1, 4, 2, 1, 2, 2, 1, 3, 6, 2, 1, 3, 2, 2, 1, 7, 3, 8, 2, 1, 4, 3, 5, 2, 1, 4, 9, 3, 5, 2, 1, 4, 6, 3, 9, 2, 1, 10, 4, 6, 3, 15, 2, 1, 2, 4, 10, 3, 3, 2, 1, 7, 15, 4, 20, 3, 8, 2, 1, 11, 7, 4, 5, 3, 6, 2, 1, 5, 11, 7, 4, 14, 3, 6, 2, 1

Offset: 1

Clark Kimberling, Jan 27 2012

nonn

For a guide to related sequences, see A204892.

Cf. A205402, A204892.

Mathematica
```
(See the program at A205402.)
```

Definition corrected by Clark Kimberling, Dec 05 2021

A205400 Ordered differences of quarter-squares.

Original entry on oeis.org

1, 3, 2, 5, 4, 2, 8, 7, 5, 3, 11, 10, 8, 6, 3, 15, 14, 12, 10, 7, 4, 19, 18, 16, 14, 11, 8, 4, 24, 23, 21, 19, 16, 13, 9, 5, 29, 28, 26, 24, 21, 18, 14, 10, 5, 35, 34, 32, 30, 27, 24, 20, 16, 11, 6, 41, 40, 38, 36, 33, 30, 26, 22, 17, 12, 6, 48, 47, 45, 43, 40, 37, 33

Offset: 1

Clark Kimberling, Jan 27 2012

nonn

For a guide to related sequences, see A204892.

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

Cf. A205402, A204892.

Mathematica
```
(See the program at A205402.)
```

A205404 Least s(k) such that n divides s(k)-s(j) for some j < k, where s(j) = floor((j+1)^2/2)/2.

Original entry on oeis.org

2, 4, 4, 6, 6, 12, 9, 9, 20, 12, 12, 16, 25, 16, 16, 20, 36, 20, 20, 36, 25, 42, 25, 25, 56, 30, 36, 30, 30, 36, 56, 36, 42, 36, 36, 42, 49, 42, 64, 42, 42, 72, 49, 56, 49, 110, 49, 49, 100, 56, 81, 56, 110, 56, 56, 72, 121, 64, 169, 64, 81, 64, 64, 100, 81, 72

Offset: 1

Clark Kimberling, Jan 27 2012

nonn

For a guide to related sequences, see A204892.

Cf. A205402, A204892.

Mathematica
```
(See the program at A205402.)
```

Definition corrected by Clark Kimberling, Dec 05 2021

A205405 s(A205403), where s(j)=floor[(j+1)^2/2].

Original entry on oeis.org

1, 2, 1, 2, 1, 6, 2, 1, 2, 2, 1, 4, 12, 2, 1, 4, 2, 2, 1, 16, 4, 20, 2, 1, 6, 4, 9, 2, 1, 6, 25, 4, 9, 2, 1, 6, 12, 4, 25, 2, 1, 30, 6, 12, 4, 64, 2, 1, 2, 6, 30, 4, 4, 2, 1, 16, 64, 6, 110, 4, 20, 2, 1, 36, 16, 6, 9, 4, 12, 2, 1, 9, 36, 16, 6, 56, 4, 12, 2, 1

Offset: 1

Clark Kimberling, Jan 27 2012

nonn

For a guide to related sequences, see A204892.

Cf. A205402, A204892.

Mathematica
```
(See the program at A205402.)
```

A205401 Least h such that n divides the h-th difference between distinct numbers quarter-squares; the differences are ordered as in A205400.

Original entry on oeis.org

1, 3, 2, 5, 4, 14, 8, 7, 23, 12, 11, 18, 34, 17, 16, 24, 47, 23, 22, 52, 31, 63, 30, 29, 82, 39, 50, 38, 37, 49, 87, 48, 60, 47, 46, 59, 72, 58, 100, 57, 56, 115, 70, 84, 69, 186, 68, 67, 155, 82, 130, 81, 174, 80, 79, 112, 205, 95, 296, 94, 128, 93, 92, 164, 127

Offset: 1

Clark Kimberling, Jan 27 2012

nonn

For a guide to related sequences, see A204892.

Cf. A205402, A204892.

Mathematica
```
(See the program at A205402.)
```

Showing 1-8 of 8 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).

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A205406 a(n) = s(k)-s(j), where (k,j) is the least pair for which n divides s(k)-s(j), and s(j) = floor((j+1)^2/2)/2.

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A198293 a(n) = (1/n)*A205406(n).

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A205403 a(n) is the index j < k such that n divides s(k)-s(j) for some j, where s(j) = floor((j+1)^2/2)/2, and k is the least index for which such a j exists.

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A205400 Ordered differences of quarter-squares.

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A205404 Least s(k) such that n divides s(k)-s(j) for some j < k, where s(j) = floor((j+1)^2/2)/2.

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A205405 s(A205403), where s(j)=floor[(j+1)^2/2].

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A205401 Least h such that n divides the h-th difference between distinct numbers quarter-squares; the differences are ordered as in A205400.

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