A204892 -id:A204892 - cp's OEIS frontend

A204894 Least prime p such that n divides p-q for some prime q

3, 5, 5, 7, 7, 11, 17, 11, 11, 13, 13, 17, 29, 17, 17, 19, 19, 23, 41, 23, 23, 29, 53, 29, 53, 29, 29, 31, 31, 37, 67, 37, 71, 37, 37, 41, 79, 41, 41, 43, 43, 47, 89, 47, 47, 53, 97, 53, 101, 53, 53, 59, 109, 59, 113, 59, 59, 61, 61, 67, 127, 67, 131, 67, 67, 71

Offset: 1

Clark Kimberling, Jan 20 2012

nonn

For a guide to related sequences, see A204892.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
Tony Haddad, Sun-Kai Leung, and Cihan Sabuncu, Visiting early at prime times, arXiv preprint (2024). arXiv:2408.11781 [math.NT]

Cf. A204892, A204903, A000040.

Mathematica
```
(See the program at A204892.)
```

a(n)=forprime(p=n+2, , forstep(k=p%n, p-1, n, if(isprime(k), return(p)))) \\ Charles R Greathouse IV, Mar 20 2013

a(n)=if(isprime(n+2),return(n+2)); my(s=if(n%2,2*n,n),t); forprime(p=s+3,, t=p%n; forstep(q=if(t%2,t,t+n),p-s,s,if(isprime(q), return(p)))) \\ Charles R Greathouse IV, Jul 17 2015

a(n)=if(isprime(n+2),return(n+2)); my(s=if(n%2,2*n,n),r); forprime(p=s+3,2*s+1, if(isprime(p-s), return(p))); forprime(p=2*s+3,, r=p%n; forstep(q=if(r%2,r,r+n),p-s,s,if(isprime(q), return(p)))) \\ Charles R Greathouse IV, Aug 31 2024

n + 2 <= a(n) <= prime(n+1). - Charles R Greathouse IV, Jul 17 2015

Haddad, Leung, & Sabuncu prove that a(n) < 270*n for all large n. Probably this holds for all n. - Charles R Greathouse IV, Aug 29 2024

A204898 Ordered differences of odd primes.

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Jan 20 2012

nonn

For a guide to related sequences, see A204892.

Cf. A204900, A204892.

Mathematica
```
(See the program at A204900.)
```

A204903 The odd prime q 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

3, 3, 5, 3, 3, 5, 3, 3, 5, 3, 7, 5, 3, 3, 7, 3, 3, 5, 3, 3, 5, 7, 7, 5, 3, 3, 5, 3, 3, 7, 5, 5, 5, 3, 3, 5, 5, 3, 5, 3, 7, 5, 3, 3, 7, 7, 3, 5, 3, 3, 5, 7, 3, 5, 3, 3, 13, 3, 13, 7, 5, 5, 5, 3, 7, 5, 3, 3, 11, 3, 7, 7, 3, 5, 7, 3, 3, 5, 5, 3, 5, 7, 7, 5, 3, 3, 5, 13, 3, 7

Offset: 1

Clark Kimberling, Jan 20 2012

nonn

For a guide to related sequences, see A204892.

Cf. A204900, A204892.

Mathematica
```
(See the program at A204900.)
```

A204905 Least k such that n divides k^2-j^2 for some j satisfying 1<=j

Original entry on oeis.org

2, 3, 2, 3, 3, 4, 4, 3, 5, 6, 6, 4, 7, 8, 4, 5, 9, 9, 10, 6, 5, 12, 12, 5, 10, 14, 6, 8, 15, 8, 16, 6, 7, 18, 6, 9, 19, 20, 8, 7, 21, 10, 22, 12, 7, 24, 24, 7, 14, 15, 10, 14, 27, 12, 8, 9, 11, 30, 30, 8

Offset: 1

Clark Kimberling, Jan 21 2012

nonn

See A204892 for a discussion and guide to related sequences.

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

Cf. A000290, A204892.

s[n_] := s[n] = n^2; z1 = 600; z2 = 60;
Table[s[n], {n, 1, 30}]     (* A000290 *)
u[m_] := u[m] = Flatten[Table[s[k] - s[j], {k, 2, z1}, {j, 1, k - 1}]][[m]]
Table[u[m], {m, 1, z1}]     (* A120070 *)
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}]   (* A204994 *)
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}]       (* A204905 *)
Table[j[n], {n, 1, z2}]       (* A204995 *)
Table[s[k[n]], {n, 1, z2}]    (* A204996 *)
Table[s[j[n]], {n, 1, z2}]    (* A204997 *)
Table[s[k[n]] - s[j[n]], {n, 1, z2}]     (* A204998 *)
Table[(s[k[n]] - s[j[n]])/n, {n, 1, z2}] (* A204999 *)

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

A204926 Least Fibonacci number f such that n divides f-g for some Fibonacci number g satisfying g < f.

Original entry on oeis.org

2, 3, 5, 5, 8, 8, 8, 13, 21, 13, 13, 13, 21, 55, 233, 21, 55, 21, 21, 21, 34, 89, 233, 233, 55, 34, 55, 89, 34, 2584, 34, 34, 34, 55, 17711, 233, 610, 89, 4181, 2584, 144, 55, 89, 89, 233, 233, 55, 233, 377, 55, 2584, 55, 55, 55, 89, 2584, 233, 233, 121393

Offset: 1

Clark Kimberling, Jan 21 2012

nonn

For a guide to related sequences, see A204892.

G. C. Greubel, Table of n, a(n) for n = 1..1000

Cf. A204924, A204892, A000045.

Mathematica
```
(See the program at A204924.)
```

A204927 The number s(j) such that n divides s(k)-s(j), where s(j) is the (j+1)-st Fibonacci number and k is the least positive integer for which such a j>0 exists.

Original entry on oeis.org

1, 1, 2, 1, 3, 2, 1, 5, 3, 3, 2, 1, 8, 13, 8, 5, 21, 3, 2, 1, 13, 1, 3, 89, 5, 8, 1, 5, 5, 34, 3, 2, 1, 21, 1, 89, 55, 13, 8, 144, 21, 13, 3, 1, 8, 3, 8, 89, 34, 5, 34, 3, 2, 1, 34, 8, 5, 1, 89, 1597

Offset: 1

Clark Kimberling, Jan 21 2012

nonn

For a guide to related sequences, see A204892.

G. C. Greubel, Table of n, a(n) for n = 1..1000

Cf. A204924, A204925, A204892, A000045.

Mathematica
```
(See the program at A204924.)
```

A204933 The index jA204932) for which such j exists.

Original entry on oeis.org

1, 2, 3, 2, 1, 3, 1, 4, 3, 5, 2, 4, 4, 3, 5, 4, 1, 3, 3, 5, 3, 2, 1, 4, 5, 4, 6, 7, 4, 5, 2, 4, 4, 3, 7, 6, 6, 3, 4, 5, 5, 3, 8, 4, 6, 4, 4, 4, 7, 5, 3, 4, 2, 6, 6, 7, 3, 4, 2, 5, 8, 2, 7, 8, 13, 4, 5, 5, 4, 7, 7, 6, 4, 6, 5, 4, 11, 4, 9, 6, 9, 5, 3, 7, 5, 8, 4, 4, 8, 6, 13, 4, 11, 4, 13, 4

Offset: 1

Clark Kimberling, Jan 21 2012

nonn

For a guide to related sequences, see A204892.

Cf. A204932, A204892, A204935.

Mathematica
```
(See the program at A204932.)
```

A204937 (k(n)!-j(n)!)/n, where (k!,j!) is the least pair of distinct factorials for which n divides k!-j!.

Original entry on oeis.org

1, 2, 6, 1, 1, 3, 17, 12, 2, 60, 2, 8, 27912, 51, 40, 6, 7, 1, 6, 30, 34, 1, 1, 4, 24, 13956, 160, 1260, 24, 20, 117058, 3, 152, 21, 1008, 120, 168297840, 3, 9304, 15, 120, 17, 927360, 114, 96, 876, 77208, 2, 720, 12, 14, 6978, 9037766, 80, 720, 630, 2

Offset: 1

Clark Kimberling, Jan 21 2012

nonn

The ordering of differences k!-j! is given at A204930. For a guide to related sequences, see A204892.

Cf. A204932, A204892.

Mathematica
```
(See the program at A204932.)
```

A204983 a(n) = 2^(k-1)-2^(j-1), where (2^(k-1),2^(j-1)) is the least pair of distinct positive powers of 2 for which n divides 2^(k-1)-2^(j-1).

Original entry on oeis.org

1, 2, 3, 4, 15, 6, 7, 8, 63, 30, 1023, 12, 4095, 14, 15, 16, 255, 126, 262143, 60, 63, 2046, 2047, 24, 1048575, 8190, 262143, 28, 268435455, 30, 31, 32, 1023, 510, 4095, 252, 68719476735, 524286, 4095, 120, 1048575, 126, 16383, 4092, 4095

Offset: 1

Clark Kimberling, Jan 21 2012

nonn

For a guide to related sequences, see A204892.

(Conjecture) Equivalently, the solution set of 2^p * (2^q - 1) = x * y, OR 2^q - 1 = 2^p * x * y, for at most one of the naturals x and y being given; unknown p and q in the integers; then a(n) = 2^p * (2^q - 1) where p and q are directly related to n (see formula). - Andrew T. Porter, Dec 20 2022

Cf. A204979, A204892, A204984.
Cf. A000079, A173787.
Cf. A007733, A007814.

Mathematica
```
(See the program at A204979.)
```

a(n) = for (k=1, oo, for (j=1, k-1, my(d=2^(k-1)-2^(j-1)); if (!(d % n), return(d)););); \\ Michel Marcus, Sep 16 2023

Conjecture: a(n) = 2^A007814(n) * (2^A007733(n) - 1). - Andrew T. Porter, Dec 20 2022

cp's OEIS Frontend

A204894 Least prime p such that n divides p-q for some prime q

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A204898 Ordered differences of odd primes.

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A204903 The odd prime q such that n divides p-q, where p>q is the least prime for which such a prime q exists.

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A204905 Least k such that n divides k^2-j^2 for some j satisfying 1<=j

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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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Maple

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A204926 Least Fibonacci number f such that n divides f-g for some Fibonacci number g satisfying g < f.

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Mathematica

A204927 The number s(j) such that n divides s(k)-s(j), where s(j) is the (j+1)-st Fibonacci number and k is the least positive integer for which such a j>0 exists.

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Mathematica

A204933 The index jA204932) for which such j exists.

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Mathematica

A204937 (k(n)!-j(n)!)/n, where (k!,j!) is the least pair of distinct factorials for which n divides k!-j!.

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