A205146 -id:A205146 - 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

A205150 s(k)-s(j), where (s(k),s(j)) is the least such pair for which n divides their difference, and s(j)=prime(j)*prime(j+1).

Original entry on oeis.org

9, 20, 9, 20, 20, 42, 42, 128, 9, 20, 66, 108, 78, 42, 180, 128, 102, 108, 114, 20, 42, 66, 230, 144, 850, 78, 108, 308, 29, 180, 62, 128, 66, 102, 2485, 108, 370, 114, 78, 360, 246, 42, 215, 308, 180, 230, 893, 144, 294, 850, 102, 884, 636, 108, 1980

Offset: 1

Clark Kimberling, Jan 25 2012

nonn

For a guide to related sequences, see A204892.

Michel Marcus, Table of n, a(n) for n = 1..10000

Cf. A205146, A204892.

Mathematica
```
(See the program at A205146.)
```

s(m) = prime(m)*prime(m+1);
isok(k, n) = my(sk=s(k)); for (j=1, k-1, if (!Mod(sk-s(j), n), return (j)));
a(n) = my(k=1, x); while (!(j=isok(k, n)), k++); s(k) - s(j); \\ Michel Marcus, Jul 23 2021

A205151 a(n) = A205150(n)/n.

Original entry on oeis.org

9, 10, 3, 5, 4, 7, 6, 16, 1, 2, 6, 9, 6, 3, 12, 8, 6, 6, 6, 1, 2, 3, 10, 6, 34, 3, 4, 11, 1, 6, 2, 4, 2, 3, 71, 3, 10, 3, 2, 9, 6, 1, 5, 7, 4, 5, 19, 3, 6, 17, 2, 17, 12, 2, 36, 11, 2, 4, 10, 3, 50, 1, 12, 2, 31, 1, 6, 13, 74, 23, 1, 2, 50, 5, 24, 23, 4, 1, 8

Offset: 1

Clark Kimberling, Jan 25 2012

nonn

For a guide to related sequences, see A204892.

Michel Marcus, Table of n, a(n) for n = 1..10000

Cf. A205146, A204892.

Mathematica
```
(See the program at A205146.)
```

s(m) = prime(m)*prime(m+1);
isok(k, n) = my(sk=s(k)); for (j=1, k-1, if (!Mod(sk-s(j), n), return (j)));
a(n) = my(k=1, x); while (!(j=isok(k, n)), k++); (s(k) - s(j))/n; \\ Michel Marcus, Jul 23 2021

More terms from Michel Marcus, Jul 23 2021

A205144 Ordered differences of distinct binary products of consecutive primes.

Original entry on oeis.org

9, 29, 20, 71, 62, 42, 137, 128, 108, 66, 215, 206, 186, 144, 78, 317, 308, 288, 246, 180, 102, 431, 422, 402, 360, 294, 216, 114, 661, 652, 632, 590, 524, 446, 344, 230, 893, 884, 864, 822, 756, 678, 576, 462, 232, 1141, 1132, 1112, 1070, 1004

Offset: 1

Clark Kimberling, Jan 25 2012

nonn

For a guide to related sequences, see A204892.

			a(1)=s(2)-s(1)=15-6=9
a(2)=s(3)-s(1)=35-6=29
a(3)=s(3)-s(2)=35-15=20
a(4)=s(4)-s(1)=77-6=71
a(5)=s(4)-s(2)=77-15=62

Cf. A205146, A204892.

Mathematica
```
(See the program at A205146.)
```

A205148 Least s(k) such that n divides s(k)-s(j) for some j

Original entry on oeis.org

15, 35, 15, 35, 35, 77, 77, 143, 15, 35, 143, 143, 221, 77, 323, 143, 323, 143, 437, 35, 77, 143, 667, 221, 1517, 221, 143, 323, 35, 323, 77, 143, 143, 323, 2491, 143, 1517, 437, 221, 437, 323, 77, 221, 323, 323, 667, 899, 221, 437, 1517, 323, 899

Offset: 1

Clark Kimberling, Jan 25 2012

nonn

For a guide to related sequences, see A204892.

Cf. A205146, A204892.

Mathematica
```
(See the program at A205146.)
```

A205149 The number s(j) such that n divides s(k)-s(j), where s(j)=prime(j)*prime(j+1) and k is the least positive integer for which such a j exists.

Original entry on oeis.org

6, 15, 6, 15, 15, 35, 35, 15, 6, 15, 77, 35, 143, 35, 143, 15, 221, 35, 323, 15, 35, 77, 437, 77, 667, 143, 35, 15, 6, 143, 15, 15, 77, 221, 6, 35, 1147, 323, 143, 77, 77, 35, 6, 15, 143, 437, 6, 77, 143, 667, 221, 15, 2491, 35, 1147, 1147, 323, 667, 77

Offset: 1

Clark Kimberling, Jan 25 2012

nonn

For a guide to related sequences, see A204892.

Cf. A205146, A204892.

Mathematica
```
(See the program at A205146.)
```

A205145 Least k such that n divides a difference between distinct binary products of consecutive primes, as ordered in A205144.

Original entry on oeis.org

1, 3, 1, 3, 3, 6, 6, 8, 1, 3, 10, 9, 15, 6, 20, 8, 21, 9, 28, 3, 6, 10, 36, 14, 64, 15, 9, 17, 2, 20, 5, 8, 10, 21, 92, 9, 66, 28, 15, 25, 19, 6, 11, 17, 20, 36, 37, 14, 26, 64, 21, 38, 120, 9, 116, 77, 28, 45, 32, 20

Offset: 1

Clark Kimberling, Jan 25 2012

nonn

For a guide to related sequences, see A204892.

Cf. A205146, A204892.

Mathematica
```
(See the program at A205146.)
```

Showing 1-8 of 8 results.

cp's OEIS Frontend

A205147 The index jA205146) for which such j exists, and s(k)=prime(k)*prime(k+1).

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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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A205150 s(k)-s(j), where (s(k),s(j)) is the least such pair for which n divides their difference, and s(j)=prime(j)*prime(j+1).

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A205151 a(n) = A205150(n)/n.

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A205144 Ordered differences of distinct binary products of consecutive primes.

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A205148 Least s(k) such that n divides s(k)-s(j) for some j

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A205149 The number s(j) such that n divides s(k)-s(j), where s(j)=prime(j)*prime(j+1) and k is the least positive integer for which such a j exists.

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A205145 Least k such that n divides a difference between distinct binary products of consecutive primes, as ordered in A205144.

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