A024898 -id:A024898 - cp's OEIS frontend

A333721 Numbers k such that k + 1, 2k + 1, 3k + 1, 4k + 1, and 6k + 1 are all prime.

1530, 4260, 25410, 26040, 78540, 111720, 174990, 211050, 214830, 395430, 403260, 409290, 459690, 487830, 512820, 711120, 779790, 910560, 1023750, 1135950, 1280370, 1312350, 1451520, 1464810, 1487070, 1563510, 1623360, 1698060, 1824330, 1933680, 2006340, 2097480

Offset: 1

Pedro Caceres, May 04 2020

nonn

All terms are multiples of 6.

All terms are multiples of 30. - Robert Israel, Jun 17 2020

			25410 is in the sequence because 25411, 50821, 76231, 101641, 152461 are all prime.

Robert Israel, Table of n, a(n) for n = 1..10000
Pedro Caceres, 30 Ideas about Prime Numbers

Cf. A006093, A005097, A024892, A087370, A005098, A005099, A024898, A024899.

select(t -> andmap(isprime, [t+1,2*t+1,3*t+1,4*t+1,6*t+1]), [seq(i,i=30..3*10^6,30)]); # Robert Israel, Jun 17 2020

isok(m)={for(i=1, 6, if(i<>5&&!isprime(i*m+1), return(0))); 1}
{ forstep(n=0, 3*10^6, 6, if(isok(n), print1(n, ", "))) } \\ Andrew Howroyd, May 04 2020

A280314 Relationship of prime numbers to multiples of 6. The value of a(n) = 2 if n6 has two neighboring primes, a(n) = 1 if only n6+1 is prime, a(n) = -1 if only n6-1 is prime, and a(n) = 0 if the neighbors of n6 are both composite.

Original entry on oeis.org

2, 2, 2, -1, 2, 1, 2, -1, -1, 2, 1, 2, 1, -1, -1, 1, 2, 2, -1, 0, 1, -1, 2, 0, 2, 1, 1, -1, -1, 2, 0, 2, 2, 0, 1, 0, 1, 2, -1, 2, 0, -1, -1, -1, 2, 1, 2, 0, -1, 0, 1, 2, -1, 0, 1, 1, 0, 2, -1, -1, 1, 1, 1, -1, -1, 1, -1, 1, 0, 2, 0, 2, 1, -1, -1, 1, 2, -1, 0, -1, 1, -1, 1, -1, -1, 0, 2, 0, 0, 1, 1

Offset: 1

Shannon Jacobs, Dec 31 2016

sign

The number of 2's will decrease and the number of 0's will increase as n increases. If there is any pattern (even a local pattern), then the sequence will generate prime numbers, so I predict the values of the sequence have no pattern.

			From _Michael De Vlieger_, Dec 31 2016: (Start)
a(1) = 2 since both 1(6)-1 = 5 and 1(6)+1 = 7 are prime.
a(4) = -1 since only 4(6)-1 = 23 is prime; 4(6)+1 = 25 is divisible by 5.
a(20) = 0 since neither 20(6)-1 = 119 nor 20(6)+1 = 121 are prime.
(End)

Cf. A024898, A024899, A007528, A002476.

Table[If[Times @@ Abs@ # == 1, Total@ Abs@ #, Total@ #] &[{-1, 1} Boole@ Map[PrimeQ, n + {-1, 1}]], {n, 6, 546, 6}] (* Michael De Vlieger, Dec 31 2016 *)

A330410 a(n) = 6*prime(n) - 1.

Original entry on oeis.org

11, 17, 29, 41, 65, 77, 101, 113, 137, 173, 185, 221, 245, 257, 281, 317, 353, 365, 401, 425, 437, 473, 497, 533, 581, 605, 617, 641, 653, 677, 761, 785, 821, 833, 893, 905, 941, 977, 1001, 1037, 1073, 1085, 1145, 1157, 1181, 1193, 1265, 1337, 1361, 1373, 1397, 1433, 1445

Offset: 1

M. F. Hasler, Dec 13 2019

nonn

Composite terms are a(k) with k in {5, 6, 11, 12, 13, 18, 20, 21, ...} = indices of primes missing in A158015. Primes are A016969(A158015 - 1).

Cf. A000040 (primes), A016969 (6n+5), A024898 (6n-1 is prime), A158015 (primes in A024898), A049452 = {n(6n-1)}, A255584 = A033570(A130800) (semiprimes (2n+1)(3n+1)), A245365 (primes of the form n(3n-1)/2).

PARI
```
apply( a(n)=6*prime(n)-1, [1..99])
```
PARI
```
apply( n->6*n-1, primes(99))
```

a(n) = A016969(A000040(n)-1) = 6p - 1 with p = A000040(n) = prime(n).

A377540 Numbers k such that at least one of the numbers 6k-1 or 6k+1 is prime.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 21, 22, 23, 25, 26, 27, 28, 29, 30, 32, 33, 35, 37, 38, 39, 40, 42, 43, 44, 45, 46, 47, 49, 51, 52, 53, 55, 56, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 70, 72, 73, 74, 75, 76, 77, 78, 80, 81, 82, 83, 84, 85, 87

Offset: 1

Michel Eduardo Beleza Yamagishi, Oct 31 2024

Union of A024898 and A024899.
Cf. A002476, A007528.
Complement of A060461 (with respect to the positive integers) or A171696 (with respect to the nonnegative integers).

Select[Range[100], PrimeQ[6 # - 1] || PrimeQ[6 # + 1] &]

isok(k) = isprime(6*k-1) || isprime(6*k+1); \\ Michel Marcus, Oct 31 2024

A382393 Positive integers k such that 6k - 1 is prime for k != 1 (mod 5) and (6k - 1)/5 is prime for k == 1 (mod 5).

Original entry on oeis.org

2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 15, 16, 17, 18, 19, 22, 23, 25, 26, 28, 29, 30, 31, 32, 33, 36, 38, 39, 40, 42, 43, 44, 45, 47, 49, 51, 52, 53, 56, 58, 59, 60, 61, 64, 65, 66, 67, 70, 72, 74, 75, 77, 78, 80, 81, 82, 84, 85, 86, 87, 91, 93, 94, 95, 98, 99, 100

Offset: 1

V. Barbera, Mar 23 2025

nonn

For all elements of this sequence, there are no pairs (x,y) of positive integers with x > 1 such that a(n)=6*x*y+x-y.

Cf. A024898, A024899.

select(k->isprime((6*k-1)/(5-4*!(k==Mod(1,5)))), vector(100,i,i))

cp's OEIS Frontend

A333721 Numbers k such that k + 1, 2k + 1, 3k + 1, 4k + 1, and 6k + 1 are all prime.

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A280314 Relationship of prime numbers to multiples of 6. The value of a(n) = 2 if n*6 has two neighboring primes, a(n) = 1 if only n*6+1 is prime, a(n) = -1 if only n*6-1 is prime, and a(n) = 0 if the neighbors of n*6 are both composite.

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A330410 a(n) = 6*prime(n) - 1.

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A377540 Numbers k such that at least one of the numbers 6k-1 or 6k+1 is prime.

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Mathematica

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A382393 Positive integers k such that 6*k - 1 is prime for k != 1 (mod 5) and (6*k - 1)/5 is prime for k == 1 (mod 5).

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PARI

A280314 Relationship of prime numbers to multiples of 6. The value of a(n) = 2 if n6 has two neighboring primes, a(n) = 1 if only n6+1 is prime, a(n) = -1 if only n6-1 is prime, and a(n) = 0 if the neighbors of n6 are both composite.

A382393 Positive integers k such that 6k - 1 is prime for k != 1 (mod 5) and (6k - 1)/5 is prime for k == 1 (mod 5).