A085063 -id:A085063 - cp's OEIS frontend

A120223 a(n) is the minimal number k>1 such that n+k and n*k+1 are primes.

2, 3, 2, 3, 2, 5, 4, 5, 2, 3, 2, 5, 4, 3, 2, 7, 6, 11, 10, 3, 2, 9, 6, 13, 4, 3, 4, 15, 2, 7, 10, 11, 10, 3, 2, 5, 4, 5, 2, 7, 2, 5, 4, 9, 14, 13, 6, 5, 4, 3, 2, 21, 14, 5, 6, 5, 4, 9, 12, 7, 6, 5, 10, 3, 2, 5, 4, 15, 2, 3, 8, 25, 6, 27, 8, 3, 6, 11, 4, 3, 2, 15, 6, 5, 12, 11, 20, 15, 12, 7, 6, 5, 4, 3

Offset: 1

Zak Seidov, Jun 10 2006

nonn

If n+1 is prime then a(n)>A085063(n); if n+1 is not prime then a(n)=A085063(n).

			a(3)=2 because 3+2=5 and 3*2+1=7 are prime;
a(8)=5 because 8+5=13 and 8*5+1=41 are prime.

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

Cf. A085063, A092945, A085063, A120224, A120225.

f:= proc(n) local k;
  for k from `if`( n::odd,2,3) do
     if isprime(n*k+1) and isprime(n+k) then return k fi
  od
end proc:
map(f, [$1..100]); # Robert Israel, Feb 03 2019

Reap[Do[Do[If[PrimeQ[{n+x, n*x+1}]=={True,True},Sow[x];Break[]],{x,2,100}],{n,120}]][[2,1]]
mnk[n_]:=Module[{k=2},While[!AllTrue[{n+k,n*k+1},PrimeQ],k++];k]; Array[ mnk,100] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Oct 15 2014 *)

for(n=1,100,k=2;while(!isprime(n+k),k++;while(!isprime(n*k+1),k++));print1(k,", ")) \\ Jinyuan Wang, Feb 04 2019

A120224 a(n) is the minimal number k>=n such that n+k and n*k+1 are primes.

Original entry on oeis.org

1, 3, 4, 7, 6, 7, 10, 9, 14, 13, 18, 19, 24, 15, 16, 21, 24, 29, 22, 21, 22, 31, 30, 43, 28, 33, 34, 39, 32, 41, 36, 39, 34, 37, 66, 43, 60, 41, 50, 43, 42, 55, 46, 53, 52, 51, 50, 59, 52, 51, 56, 55, 56, 55, 58, 75, 74, 69, 68, 67, 66, 75, 74, 67, 86, 83, 70, 89, 70, 79, 102, 79

Offset: 1

Zak Seidov, Jun 10 2006

nonn

Differs from A120225 in the first term.

			a(3)=4 because 3+4=7 and 3*4+1=13 are prime;
a(8)=9 because 8+9=17 and 8*9+1=73 are prime,

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A085063, A092945, A085063, A120223, A120225.

Reap[Do[Do[If[PrimeQ[{n+x, n*x+1}]=={True,True},Sow[x];Break[]],{x, n,1000}],{n,120}]][[2,1]]
mnk[n_]:=Module[{k=n},Until[AllTrue[{n+k,n*k+1},PrimeQ],k++];k]; Join[{1},Array[mnk,80,2]] (* Harvey P. Dale, May 12 2025 *)

A120225 a(n) is the minimal number k>n such that n+k and n*k+1 are primes.

Original entry on oeis.org

2, 3, 4, 7, 6, 7, 10, 9, 14, 13, 18, 19, 24, 15, 16, 21, 24, 29, 22, 21, 22, 31, 30, 43, 28, 33, 34, 39, 32, 41, 36, 39, 34, 37, 66, 43, 60, 41, 50, 43, 42, 55, 46, 53, 52, 51, 50, 59, 52, 51, 56, 55, 56, 55, 58, 75, 74, 69, 68, 67, 66, 75, 74, 67, 86, 83, 70, 89, 70, 79, 102, 79

Offset: 1

Zak Seidov, Jun 10 2006

nonn

Differs from A120224 in the first term.

			a(3)=4 because 3+4=7 and 3*4+1=13 are prime;
a(8)=9 because 8+9=17 and 8*9+1=73 are prime,

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A085063, A092945, A085063, A120223, A120224.

Reap[Do[Do[If[PrimeQ[{n+x, n*x+1}]=={True,True},Sow[x];Break[]],{x, n+1,1000}],{n,120}]][[2,1]]
mnk[n_]:=Module[{k=n+1},While[!AllTrue[{n+k,n*k+1},PrimeQ],k++];k]; Array[ mnk,80] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, May 15 2015 *)

A120226 Numbers k such that 4+k and 4*k+1 are prime.

Original entry on oeis.org

1, 3, 7, 9, 13, 15, 25, 27, 37, 39, 43, 49, 57, 67, 69, 79, 93, 97, 99, 105, 127, 135, 153, 163, 169, 175, 177, 189, 193, 207, 219, 235, 253, 265, 267, 273, 277, 279, 303, 307, 309, 343, 345, 363, 405, 417, 427, 435, 475, 483, 487, 499, 517, 553, 559, 567, 573

Offset: 1

Zak Seidov, Jun 10 2006

nonn

Except for 3, no terms == 3 or 5 (mod 7). - Robert Israel, Jul 26 2019

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

Cf. A092945, A106057, A085063, A120223, A120224, A120225, A120227.

select(t -> isprime(t+4) and isprime(4*t+1), [seq(i,i=1..1000,2)]); # Robert Israel, Jul 26 2019

Select[Range[600],AllTrue[{4+#,4#+1},PrimeQ]&] (* Harvey P. Dale, Nov 14 2022 *)

A120227 Numbers k such that 5+k and 5*k+1 are prime.

Original entry on oeis.org

2, 6, 8, 12, 14, 26, 36, 38, 42, 48, 54, 56, 62, 66, 84, 92, 98, 104, 108, 126, 132, 152, 162, 176, 188, 194, 206, 218, 234, 236, 246, 258, 264, 272, 276, 302, 306, 344, 348, 362, 374, 416, 426, 428, 444, 456, 462, 474, 482, 504, 518, 542, 558, 572, 594, 602

Offset: 1

Zak Seidov, Jun 10 2006

nonn

This sequence is infinite under Dickson's conjecture. - Charles R Greathouse IV, Apr 16 2012

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A092945, A106057, A085063, A120223, A120224, A120225, A120226.

Select[Range[700],And@@PrimeQ[{#+5,5#+1}]&] (* Harvey P. Dale, Dec 18 2011 *)

A085064 a(n) = smallest k such that nk-1 as well as n+k are primes.

Original entry on oeis.org

4, 3, 2, 1, 6, 1, 6, 3, 2, 3, 12, 1, 18, 3, 2, 3, 6, 1, 12, 3, 2, 9, 6, 7, 6, 15, 2, 3, 12, 1, 30, 15, 4, 3, 12, 5, 24, 3, 8, 21, 12, 1, 18, 3, 2, 57, 6, 5, 12, 3, 2, 9, 6, 5, 12, 3, 2, 3, 42, 1, 18, 9, 4, 3, 6, 7, 6, 15, 2, 27, 18, 1, 66, 15, 8, 3, 6, 5, 120, 3, 8, 15, 18, 5, 12, 3, 2, 15, 18

Offset: 1

Amarnath Murthy, Jun 28 2003

nonn

a(n)=1 if and only if n is in A014574. - Robert Israel, May 14 2018

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

Cf. A014574, A085063.

f:= proc(n) local k;
for k from 1+(n mod 2) by 2 do
  if isprime(n+k) and isprime(n*k-1) then return k fi
od
end proc:
map(f, [$1..100]); # Robert Israel, May 14 2018

More terms from Robert Israel, May 14 2018

cp's OEIS Frontend

A120223 a(n) is the minimal number k>1 such that n+k and n*k+1 are primes.

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A120224 a(n) is the minimal number k>=n such that n+k and n*k+1 are primes.

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A120225 a(n) is the minimal number k>n such that n+k and n*k+1 are primes.

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A120226 Numbers k such that 4+k and 4*k+1 are prime.

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A120227 Numbers k such that 5+k and 5*k+1 are prime.

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A085064 a(n) = smallest k such that nk-1 as well as n+k are primes.

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