A092945 -id:A092945 - cp's OEIS frontend

A085063 a(n) is the minimal number k such that n+k and n*k+1 are primes.

1, 1, 2, 1, 2, 1, 4, 5, 2, 1, 2, 1, 4, 3, 2, 1, 6, 1, 10, 3, 2, 1, 6, 13, 4, 3, 4, 1, 2, 1, 10, 11, 10, 3, 2, 1, 4, 5, 2, 1, 2, 1, 4, 9, 14, 1, 6, 5, 4, 3, 2, 1, 14, 5, 6, 5, 4, 1, 12, 1, 6, 5, 10, 3, 2, 1, 4, 15, 2, 1, 8, 1, 6, 27, 8, 3, 6, 1, 4, 3, 2, 1, 6, 5, 12, 11, 20, 1, 12, 7, 6, 5, 4, 3, 2, 1, 4, 5

Offset: 1

Amarnath Murthy, Jun 28 2003

nonn

If n+1 is prime then a(n)=1; if n+1 is not prime then a(n)=A120223(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. A092945, A120223, A120224, A120225.

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:
f(1):= 1: # Robert Israel, May 14 2018

Reap[Do[Do[If[PrimeQ[{n+x, n*x+1}]=={True,True},Sow[x];Break[]],{x,1,100}],{n,120}]][[2,1]]
nkp[n_]:=Module[{k=1},While[!And@@PrimeQ[{n+k,n*k+1}],k++];k]; Array[nkp, 100] (* Harvey P. Dale, Apr 11 2012 *)

a(n) = {my(k=1); while (!isprime(n+k) || !isprime(n*k+1), k++); k;} \\ Michel Marcus, May 14 2018

Corrected and extended by Zak Seidov, Jun 10 2006

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

Original entry on oeis.org

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 *)

A092944 Group the natural numbers so that the n-th group contains n numbers whose sum as well as the group product + 1 is prime. Sequence contains the first member of the groups.

Original entry on oeis.org

2, 1, 3, 5, 11, 15, 20, 27, 36, 44, 54, 64, 76, 89, 102, 117, 132, 149, 166, 184, 204, 228, 249, 272, 296, 323, 349, 376, 403, 432, 461, 493, 524, 556, 589, 625, 660, 697, 737, 775, 814, 855, 898, 943, 987, 1031, 1076, 1122, 1169, 1219, 1269, 1321, 1373, 1427

Offset: 1

Amarnath Murthy, Mar 23 2004

The n-th group is chosen so that it is lexicographically earliest.
First n-2 numbers are the least previously unused numbers.
The (n-1)st number is chosen with the additional condition that if the product is odd, the sum is also odd (to avoid an impossible situation in picking the n-th number).
The n-th number is chosen as the least unused number that meets the two prime conditions.
In the 3rd group, 6 is selected as the 2nd number rather than 5, else no 3rd number could be found to meet the prime conditions.

			Table begins:
2
1,4
3,6,10
5,7,8,9
11,12,13,14,23
15,16,17,18,19,28

Cf. A092945, A092946, A092947.

Edited and extended by Ray Chandler, May 07 2008

A092946 Group the natural numbers so that the n-th group contains n numbers whose sum as well as the group product +1 is prime. Sequence contains the primes arising as the sum of the terms of groups.

Original entry on oeis.org

2, 5, 19, 29, 73, 113, 167, 269, 431, 509, 673, 977, 1193, 1423, 1861, 1993, 2467, 3041, 3391, 4003, 4523, 5309, 6011, 6833, 7993, 9239, 10093, 10909, 12157, 13417, 15199, 16651, 17971, 19477, 21517, 23197, 25121, 27799, 29537, 31891, 34583, 37189

Offset: 1

Amarnath Murthy, Mar 23 2004

nonn

See A092944 for additional clarification of definition.

Cf. A092944, A092945, A092947.

Edited and extended by Ray Chandler, May 07 2008

A092947 Group the natural numbers so that the n-th group contains n numbers whose sum as well as the group product +1 is prime. Sequence contains the primes arising as the product of the terms of groups +1.

Original entry on oeis.org

3, 5, 181, 2521, 552553, 39070081, 4180176001, 1483048828801, 672375473078401, 106890247271808001, 40812700642879334401, 37716399337002946560001, 18266163370859189769984001, 9553876078552184850831360001

Offset: 1

Amarnath Murthy, Mar 23 2004

nonn

See A092944 for additional clarification of definition.

Cf. A092944, A092945, A092946.

Edited, corrected and extended by Ray Chandler, May 07 2008

cp's OEIS Frontend

A085063 a(n) is the minimal number k such that n+k and n*k+1 are primes.

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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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Mathematica

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

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Programs

Maple

Mathematica

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

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Crossrefs

Programs

Mathematica

A092944 Group the natural numbers so that the n-th group contains n numbers whose sum as well as the group product + 1 is prime. Sequence contains the first member of the groups.

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Author

Keywords

Comments

Examples

Crossrefs

Extensions

A092946 Group the natural numbers so that the n-th group contains n numbers whose sum as well as the group product +1 is prime. Sequence contains the primes arising as the sum of the terms of groups.

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Author

Keywords

Comments

Crossrefs

Extensions