A152091 -id:A152091 - cp's OEIS frontend

A153090 Least k(n) such that k(n)3^n(3^n-1)+j is prime with j= -1 or 1 or both.

1, 1, 1, 1, 3, 1, 1, 14, 5, 5, 5, 1, 7, 6, 5, 7, 12, 1, 5, 1, 6, 29, 23, 20, 8, 6, 6, 9, 2, 10, 18, 19, 13, 57, 1, 1, 9, 10, 8, 5, 8, 8, 26, 5, 5, 6, 39, 41, 6, 9, 50, 6, 32, 6, 4, 8, 2, 79, 28, 23, 33, 78, 31, 35, 19, 32, 46, 7, 6, 116, 39, 7, 20, 6, 35, 8

Offset: 1

Pierre CAMI, Dec 18 2008

nonn

Sum_{n=1..k} a(n) / Sum_{n=1..k} n tends to 2*log(3)/7.

			1*3^1*(3^1-1)-1=5 prime as 7 so k(1)=1 1*3^2*(3^2-1)-1=71 prime as 73 so k(2)=1

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

Cf. A152414, A152091, A152092, A152093, A152094, A152095.

lk[n_]:=Module[{c=3^n (3^n-1),k=1},While[NoneTrue[k*c+{1,-1},PrimeQ],k++];k]; Array[lk,90] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Dec 23 2020 *)

Corrected by Harvey P. Dale, Dec 23 2020

A153092 Least k(n) such that k(n)6^n(6^n-1)+j is prime with j= -1 or 1 or both.

Original entry on oeis.org

1, 1, 1, 1, 3, 2, 3, 2, 20, 4, 5, 2, 9, 2, 27, 7, 12, 3, 3, 2, 3, 6, 6, 1, 18, 8, 2, 17, 2, 14, 55, 1, 18, 4, 59, 18, 30, 26, 32, 3, 14, 59, 42, 35, 40, 22, 7, 17, 26, 6, 28, 3, 15, 11, 6, 32, 30, 18, 14, 4, 85, 3, 1, 65, 13, 64, 7, 18, 40, 8, 68, 5, 5, 6, 107, 7, 88, 25, 6, 3, 1, 21, 8, 12, 9

Offset: 1

Pierre CAMI, Dec 18 2008

nonn

When n increases sum k(n) for i=1 to n / sum n for i=1 to n tends to 2*log(6)/9.

			1*6^1*(6^1-1)-1=29 prime as 31 so k(1)=1.

Pierre CAMI, Table of n, a(n) for n = 1..1000.

Cf. A152414, A152090, A152091, A152093, A152094, A152095.

lkn[n_]:=Module[{c=6^n (6^n-1),k=1},While[NoneTrue[k*c+{1,-1},PrimeQ],k++];k]; Array[lkn,90] (* Harvey P. Dale, Feb 29 2024 *)

A153093 Least k(n) such that k(n)7^n(7^n-1)+j is prime with j= -1 or 1 or both.

Original entry on oeis.org

1, 1, 1, 2, 9, 1, 2, 1, 1, 5, 13, 1, 6, 1, 6, 14, 4, 4, 7, 5, 6, 5, 12, 3, 6, 3, 9, 3, 20, 9, 14, 7, 43, 22, 5, 11, 4, 4, 48, 14, 8, 56, 36, 7, 83, 47, 47, 1, 2, 74, 15, 12, 22, 15, 17, 47, 20, 20, 5, 8, 12, 26, 6, 26, 55, 16, 32, 45, 24, 46, 19, 58, 6, 29, 19, 26, 83, 45, 17, 48, 6, 16, 152

Offset: 1

Pierre CAMI, Dec 18 2008

nonn

When n increases sum k(n) for i=1 to n / sum n for i=1 to n tends to log(7)/3.7.

			1*7^1*(7^1-1)-1=41 prime as 43 so k(1)=1.

Pierre CAMI, Table of n, a(n) for n = 1..1000

A152414, A152090, A152091, A152092, A152094, A152095.

A153094 Least k(n) such that k(n)m^n(m^n-1)+j is prime with j= -1 or 1 or both and 1

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 4, 1, 2, 1, 1, 1, 4, 1, 1, 1, 3, 5, 2, 1, 1, 3, 2, 2, 2, 9, 6, 1, 6, 4, 1, 1, 4, 1, 2, 2, 3, 6, 6, 5, 1, 5, 5, 1, 2, 1, 10, 3, 5, 1, 3, 2, 1, 4, 5, 2, 2, 3, 1, 1, 2, 3, 5, 5, 6, 1, 12, 3, 5, 6, 1, 2, 9, 3, 4, 1, 1, 5, 2, 3, 4, 7, 2, 2, 16, 8, 5, 1, 5, 5, 6, 3, 2, 11, 2, 2

Offset: 1

Pierre CAMI, Dec 18 2008

nonn

when n increases sum k(n) for i=1 to n / sum n for i=1 to n tends to 0.05, 0

			1*2^1*(2^1-1)+1=3 prime so k(1)=1

Pierre CAMI, Table of n, a(n) for n = 1..1000

A152414, A152090, A152091, A152092, A152093, A152095

A153095 Least m(n) such that k(n)m(n)^n(m(n)^n-1)+j is prime with j= -1 or 1 or both and least possible k(n) with 1

Original entry on oeis.org

2, 2, 3, 2, 2, 3, 3, 5, 2, 2, 2, 3, 8, 7, 5, 4, 2, 2, 4, 3, 6, 2, 2, 5, 4, 7, 6, 4, 3, 2, 2, 2, 2, 6, 3, 3, 7, 2, 2, 2, 5, 2, 2, 3, 2, 2, 2, 5, 7, 2, 9, 5, 4, 4, 2, 2, 2, 8, 7, 4, 8, 4, 6, 5, 4, 5, 5, 4, 3, 9, 2, 2, 6, 3, 8, 8, 4, 2, 4, 8, 6, 2, 2, 4, 4, 5, 6, 4, 4, 8, 4, 6, 4, 7, 3, 4, 6, 7, 5, 2

Offset: 1

Pierre CAMI, Dec 18 2008

nonn

			1*2^1*(2^1-1)+1=3 prime so m(1)=2 1*2^2*(2^2-1)-1=11 as 13 so m(2)=2 1*2^3*(2^3-1)-1=55 composite 1*2^3*(2^3-1)+1=57 composite 1*3^3*(3^3-1)-1=71 prime as 73 so m(3)=3

Pierre CAMI, Table of n, a(n) for n = 1..1000

A152414, A152090, A152091, A152092, A152093, A152094

A274465 Primes which are the sum of cousin prime pairs - 1.

Original entry on oeis.org

17, 29, 41, 89, 137, 197, 257, 389, 449, 461, 557, 617, 701, 761, 797, 881, 929, 977, 1229, 1289, 1481, 1709, 1721, 1877, 2609, 2861, 2897, 2969, 3137, 3221, 3329, 3389, 3989, 4001, 4409, 4481, 4877, 5081, 5237, 5381, 5417, 5501, 5669, 5717, 6329, 6689, 6917, 7229

Offset: 1

Debapriyay Mukhopadhyay, Jun 24 2016

nonn

Cousin primes are prime pairs that differ by 4. Any prime p in this sequence is such that p = (p-3)/2 + (p+5)/2 - 1, where (p-3)/2 and (p+5)/2 are also primes and they differ by 4.
Proper subset of A040117 (e.g., 5 isn't in the sequence). - David A. Corneth, Jun 24 2016
Intersection of A145471 and A089531. - Michel Marcus, Jun 27 2016
Subsequence of A072669. - Michel Marcus, Jun 27 2016

			17 = 7 + 11 - 1. Note that, (17-3)/2 = 7 and (17+5)/2 = 11 and 7, 11 are cousin prime pairs.
29 = 13 + 17 - 1. Note that, (29-3)/2 = 13 and (29+5)/2 = 17 and 13, 17 are cousin prime pairs.
41 = 19 + 23 - 1. Note that, (41-3)/2 = 19 and (41+5)/2 = 23 and 19, 23 are cousin prime pairs.
89 = 43 + 47 - 1. Note that, (89-3)/2 = 43 and (89+5)/2 = 47 and 43, 47 are cousin prime pairs.

John Cerkan, Table of n, a(n) for n = 1..10000

Cf. A023200, A152091, A040117, A145471, A089531, A072669.

Select[2 # + 3 &@ Select[Prime@ Range@ 512, PrimeQ[# + 4] &], PrimeQ] (* Michael De Vlieger, Jun 26 2016 *)

is(n) = isprime(n) && isprime((n-3)/2) && isprime((n+5)/2) \\ David A. Corneth, Jun 24 2016

use ntheory ":all"; say for grep{is_prime($)} map { $+$+4-1 } sieve_prime_cluster(1,5e8,4) # _Dana Jacobsen, Apr 27 2017

Showing 1-6 of 6 results.

cp's OEIS Frontend

A153090 Least k(n) such that k(n)*3^n*(3^n-1)+j is prime with j= -1 or 1 or both.

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A153092 Least k(n) such that k(n)*6^n*(6^n-1)+j is prime with j= -1 or 1 or both.

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A153093 Least k(n) such that k(n)*7^n*(7^n-1)+j is prime with j= -1 or 1 or both.

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A153094 Least k(n) such that k(n)*m^n*(m^n-1)+j is prime with j= -1 or 1 or both and 1

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A153095 Least m(n) such that k(n)*m(n)^n*(m(n)^n-1)+j is prime with j= -1 or 1 or both and least possible k(n) with 1

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A274465 Primes which are the sum of cousin prime pairs - 1.

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Perl

A153090 Least k(n) such that k(n)3^n(3^n-1)+j is prime with j= -1 or 1 or both.

A153092 Least k(n) such that k(n)6^n(6^n-1)+j is prime with j= -1 or 1 or both.

A153093 Least k(n) such that k(n)7^n(7^n-1)+j is prime with j= -1 or 1 or both.

A153094 Least k(n) such that k(n)m^n(m^n-1)+j is prime with j= -1 or 1 or both and 1

A153095 Least m(n) such that k(n)m(n)^n(m(n)^n-1)+j is prime with j= -1 or 1 or both and least possible k(n) with 1