A024899 -id:A024899 - cp's OEIS frontend

A173230 Primes p of the form 6k+1 such that 6p+1 is also prime.

7, 13, 37, 61, 73, 103, 151, 181, 241, 271, 277, 283, 313, 331, 367, 373, 397, 577, 601, 607, 661, 727, 751, 787, 853, 907, 937, 1033, 1063, 1117, 1171, 1201, 1321, 1327, 1381, 1423, 1453, 1531, 1567, 1693, 1777, 1831, 1873, 1987, 1993, 2137, 2161, 2203, 2221, 2281, 2287, 2293, 2347, 2551, 2593, 2677, 2767, 2791, 2851, 2971, 3037, 3061, 3163, 3181, 3307, 3391

Offset: 1

Zak Seidov, Nov 22 2010

nonn

Subsequence of A007693 which in turn is subsequence of A024899.

[ p: n in [0..600] | IsPrime(6*p+1) and IsPrime(p) where p is 6*n+1 ];

sp1Q[n_]:=IntegerQ[(n-1)/6]&&PrimeQ[6n+1]; Select[Prime[Range[500]],sp1Q] (* Harvey P. Dale, Apr 08 2018 *)

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 *)

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

A173230 Primes p of the form 6k+1 such that 6p+1 is also prime.

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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.

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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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PARI

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).

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PARI

cp's OEIS Frontend

A173230 Primes p of the form 6k+1 such that 6p+1 is also prime.

Views

Author

Keywords

Crossrefs

Programs

Magma

Mathematica

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

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Author

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Programs

Mathematica

PARI

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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Author

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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).