A204908 -id:A204908 - 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

A204916 Least k such that n divides s(k)-s(j) for some j in [1,k), where s(k)=(prime(k))^2.

Original entry on oeis.org

2, 3, 3, 3, 2, 4, 3, 3, 4, 4, 6, 4, 5, 5, 4, 3, 8, 5, 7, 4, 3, 7, 10, 4, 9, 8, 10, 5, 11, 6, 10, 5, 6, 9, 7, 5, 14, 11, 5, 4, 14, 7, 13, 7, 4, 10, 16, 5, 15, 11, 8, 8, 17, 11, 6, 5, 7, 13, 18, 6

Offset: 1

Clark Kimberling, Jan 20 2012

nonn

See A204892 for a discussion and guide to related sequences

Cf. A000040, A204892, A204900, A204908.

s[n_] := s[n] = Prime[n]^2; z1 = 1000; z2 = 60;
Table[s[n], {n, 1, 30}]  (* A001248 *)
u[m_] := u[m] = Flatten[Table[s[k] - s[j], {k, 2, z1}, {j, 1, k - 1}]][[m]]
Table[u[m], {m, 1, z1}]             (* A204914 *)
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}]             (* A204915 *)
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}]             (* A204916 *)
Table[j[n], {n, 1, z2}]             (* A204917 *)
Table[s[k[n]], {n, 1, z2}]          (* A204918 *)
Table[s[j[n]], {n, 1, z2}]          (* A204919 *)
Table[s[k[n]] - s[j[n]], {n, 1, z2}]     (* A204920 *)
Table[(s[k[n]] - s[j[n]])/n, {n, 1, z2}] (* A204921 *)

A204911 The prime q>=5 such that n divides p-q, where p>q is the least prime for which such a prime q exists.

Original entry on oeis.org

5, 5, 5, 7, 7, 5, 5, 5, 5, 7, 7, 5, 5, 5, 7, 7, 7, 5, 5, 11, 5, 7, 7, 5, 11, 5, 5, 13, 13, 7, 5, 5, 5, 7, 13, 5, 5, 5, 5, 7, 7, 5, 11, 17, 7, 7, 7, 5, 5, 11, 5, 7, 7, 5, 17, 5, 13, 13, 13, 7, 5, 5, 5, 7, 7, 5, 5, 5, 11, 13, 7, 7, 5, 5, 7, 7, 13, 5, 5, 17, 5, 7, 7, 5, 11, 11, 5, 13, 13, 7, 11, 5, 5

Offset: 1

Clark Kimberling, Jan 20 2012

nonn

For a guide to related sequences, see A204892.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A204908, A204900, A204892.

f:= proc(n) local V,q,r;
  V:= Array(0..n-1); q:= 4;
  do
   q:= nextprime(q);
   r:= q mod n;
   if V[r] = 0 then V[r]:= q
   else return V[r]
   fi
  od
end proc:
map(f, [$1..100]); # Robert Israel, Jul 24 2018

Mathematica
```
(See the program at A204908.)
```

from sympy import nextprime
def a(n):
    V, q = [0 for _ in range(n)], 4
    while True:
        q = nextprime(q)
        r = q%n
        if V[r] == 0: V[r] = q
        else: return int(V[r])
print([a(n) for n in range(1, 94)]) # Michael S. Branicky, Jun 25 2024 after Robert Israel

More terms from Robert G. Wilson v, Jul 24 2018

A204914 Ordered differences of squared primes.

Original entry on oeis.org

5, 21, 16, 45, 40, 24, 117, 112, 96, 72, 165, 160, 144, 120, 48, 285, 280, 264, 240, 168, 120, 357, 352, 336, 312, 240, 192, 72, 525, 520, 504, 480, 408, 360, 240, 168, 837, 832, 816, 792, 720, 672, 552, 480, 312, 957, 952, 936, 912, 840, 792, 672

Offset: 1

Clark Kimberling, Jan 20 2012

nonn

For a guide to related sequences, see A204892.

			a(1) = s(2) - s(1) =  9 - 4 =  5;
a(2) = s(3) - s(1) = 25 - 4 = 21;
a(3) = s(3) - s(2) = 25 - 9 = 16;
a(4) = s(4) - s(1) = 49 - 4 = 45.

Michael S. Branicky, Table of n, a(n) for n = 1..10000

Cf. A204908, A204900, A204892.

Mathematica
```
(See the program at A204916.)
```

from math import isqrt
from sympy import prime, primerange
def aupton(terms):
  sqps = [p*p for p in primerange(1, prime(isqrt(2*terms)+1)+1)]
  return [b-a for i, b in enumerate(sqps) for a in sqps[:i]][:terms]
print(aupton(52)) # Michael S. Branicky, May 21 2021

A204910 Least prime p such that n divides p-q for some prime q satisfying 5<=q

Original entry on oeis.org

7, 7, 11, 11, 17, 11, 19, 13, 23, 17, 29, 17, 31, 19, 37, 23, 41, 23, 43, 31, 47, 29, 53, 29, 61, 31, 59, 41, 71, 37, 67, 37, 71, 41, 83, 41, 79, 43, 83, 47, 89, 47, 97, 61, 97, 53, 101, 53, 103, 61, 107, 59, 113, 59, 127, 61, 127, 71, 131, 67, 127, 67, 131, 71, 137, 71, 139, 73, 149, 83, 149, 79

Offset: 1

Clark Kimberling, Jan 20 2012

For a guide to related sequences, see A204892.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A204908, A204900, A204892.

f:= proc(n) local p,k;
  p:= n+4;
  do
    p:= nextprime(p);
    if ormap(isprime, [seq(p-n*k,k=1..(p-5)/n)]) then return p fi
  od
end proc:
map(f, [$1..100]); # Robert Israel, Jun 27 2019

Mathematica
```
(See the program at A204908.)
```

More terms from Robert Israel, Jun 27 2019

A204906 Ordered differences of primes >=5.

Original entry on oeis.org

2, 6, 4, 8, 6, 2, 12, 10, 6, 4, 14, 12, 8, 6, 2, 18, 16, 12, 10, 6, 4, 24, 22, 18, 16, 12, 10, 6, 26, 24, 20, 18, 14, 12, 8, 2, 32, 30, 26, 24, 20, 18, 14, 8, 6, 36, 34, 30, 28, 24, 22, 18, 12, 10, 4, 38, 36, 32, 30, 26, 24, 20, 14, 12, 6, 2, 42, 40, 36, 34, 30, 28, 24

Offset: 1

Clark Kimberling, Jan 20 2012

nonn

For a guide to related sequences, see A204892.

			Ordering (as at A204892):
s(2)-s(1)=7-5=2
s(3)-s(1)=11-5=6
s(3)-s(2)=11-7=4
s(4)-s(1)=13-5=8
s(4)-s(2)=13-7=6

Cf. A204908, A204900, A204892.

Mathematica
```
(See the program at A204908.)
```

A204919 a(n) = q^2 where q is the least prime such that n divides A204916(n)^2 - q^2.

Original entry on oeis.org

4, 9, 4, 9, 4, 25, 4, 9, 4, 9, 4, 25, 4, 9, 4, 9, 4, 49, 4, 9, 4, 25, 289, 25, 4, 49, 4, 9, 4, 49, 4, 25, 4, 121, 9, 49, 961, 49, 4, 9, 4, 121, 4, 25, 4, 289, 1681, 25, 4, 361, 4, 49, 2209, 529, 4, 9, 4, 289, 4, 49

Offset: 1

Clark Kimberling, Jan 20 2012

nonn

For a guide to related sequences, see A204892.

Original name was "Least prime q^2 such that n divides p^2-q^2 for some prime p>q", which would be A089090. - Robert Israel, May 04 2019

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A089090, A204916, A204908, A204900, A204892.

N:= 100: # to get a(1)..a(N)
A:= Vector(N): count:= 0:
p:= 2: P:= 2:
for i from 1 while count < N do
  p:= nextprime(p);
  ps:= p^2;
  P:= P, p;
  for j from 1 to i while count < N do
   qs:= P[j]^2;
   S:= convert(select(t -> t <= N and A[t]=0, numtheory:-divisors(ps-qs)),list);
   A[S]:= qs;
   count:= count + nops(S);
od od:
convert(A,list); # Robert Israel, May 04 2019

Mathematica
```
(See the program at A204916.)
```

Name corrected by Robert Israel, May 04 2019

A204907 Least k such that n divides A204906(k), the k-th difference of two primes >=5.

Original entry on oeis.org

1, 1, 2, 3, 8, 2, 11, 4, 16, 8, 23, 7, 29, 11, 38, 17, 47, 16, 56, 31, 67, 23, 80, 22, 108, 29, 92, 49, 140, 38, 121, 37, 137, 47, 194, 46, 172, 56, 191, 68, 212, 67, 234, 110, 233, 80, 255, 79, 277, 108

Offset: 1

Clark Kimberling, Jan 20 2012

nonn

For a guide to related sequences, see A204892.

			a(5)=8 because 5 divides 10, which is A204906(8).
(The 8th difference of primes>=5 is s(5)-s(2)=17-7=10.)

Cf. A204908, A204900, A204892.

Mathematica
```
(See the program at A204908.)
```

A204915 Least k such that n divides A204914(k), the k-th difference of two squared primes.

Original entry on oeis.org

1, 3, 2, 3, 1, 6, 2, 3, 4, 5, 11, 6, 7, 8, 4, 3, 22, 10, 16, 5, 2, 18, 43, 6, 29, 25, 37, 8, 46, 14, 37, 9, 11, 33, 17, 10, 89, 49, 7, 5, 79, 20, 67, 18, 4, 43, 118, 9, 92, 53, 22, 25, 135, 54, 11, 8, 16, 73, 137, 14

Offset: 1

Clark Kimberling, Jan 20 2012

nonn

For a guide to related sequences, see A204892.

Cf. A204908, A204900, A204892.

Mathematica
```
(See the program at A204916.)
```

Showing 1-10 of 13 results. Next

cp's OEIS Frontend

A204909 The index jA204908) for which such j exists.

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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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A204916 Least k such that n divides s(k)-s(j) for some j in [1,k), where s(k)=(prime(k))^2.

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A204911 The prime q>=5 such that n divides p-q, where p>q is the least prime for which such a prime q exists.

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A204914 Ordered differences of squared primes.

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A204910 Least prime p such that n divides p-q for some prime q satisfying 5<=q

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A204906 Ordered differences of primes >=5.

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A204919 a(n) = q^2 where q is the least prime such that n divides A204916(n)^2 - q^2.

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