A072669 -id:A072669 - cp's OEIS frontend

A229018 Primes of the form (3x + 2)2^x - 1.

31, 223, 1279, 3276799, 14680063, 420906795007, 2357352929951743, 32326824857489154029020587706017980088319, 173918694842377447266238495093237679339055972614143

Offset: 1

K. D. Bajpai, Sep 11 2013

nonn

Also primes of the form W(n) + W(n+1) + 1 where W(n) and W(n+1) are consecutive Woodall numbers. The n-th Woodall number = n*2^n-1.

			a(2) = 223:   for x=4: R= x*2^x-1 = 4*2^4-1 = 63 and S=  (x+1)*2^(x+1)-1 = 5*2^5-1 = 159. R+S+1 = 63+159+1 = 223 which is prime.

K. D. Bajpai, Table of n, a(n) for n = 1..19
Wikipedia, Woodall number

Cf. A003261, A072669.

KD:= proc() local a,b,d;  a:= x*2^x-1;  b:=(x+1)*2^(x+1)-1;  d:=a+b+1;  if isprime(d) then   RETURN(d): fi; end: seq(KD(),x=1..1000);

Select[Table[(3*x + 2)*2^x - 1, {x, 200}], PrimeQ] (* T. D. Noe, Sep 20 2013 *)

A229075 Primes of the form p^2 + q^2 + 21, where p and q are consecutive primes.

Original entry on oeis.org

191, 311, 479, 911, 1823, 2351, 4079, 5039, 6311, 8231, 9551, 10391, 13151, 14831, 17351, 22079, 24671, 33311, 35951, 41543, 51239, 57839, 61991, 69263, 73751, 76079, 84143, 101279, 103991, 106751, 111431, 115223, 141551, 145823, 198479, 210071, 223151, 263591

Offset: 1

K. D. Bajpai, Sep 12 2013

nonn

Conjecture: the expression p^2+q^2+c with p and q consecutive primes and c=21 generates more primes than any other value of c in the range 1..150. Hence, c=21 is considered for this sequence.

			a(1) = 191: prime(4)^2 + prime(4+1)^2 + 21 = 191, which is prime.

K. D. Bajpai, Table of n, a(n) for n = 1..5000

Cf. A072669, A227340.

KD:= proc() local a; a:= ithprime(n)^2+ithprime(n+1)^2+21;  if isprime(a) then RETURN(a):  fi;  end: seq(KD(),n=1..300);

Select[Table[Prime[n]^2 + Prime[n + 1]^2 + 21, {n, 100}], PrimeQ] (* T. D. Noe, Sep 12 2013 *)

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

A224793 Least prime p which generates exactly n primes of the form p+q-1 where q < p is prime, or 0 if (conjecturally) no such p exists.

Original entry on oeis.org

2, 5, 11, 13, 47, 41, 31, 107, 43, 73, 131, 61, 191, 97, 293, 139, 353, 127, 163, 151, 0, 229, 283, 223, 659, 181, 929, 313, 241, 211, 367, 701, 271, 397, 379, 457, 337, 1031, 1259, 607, 331, 463, 643, 613, 1409, 733, 911, 1091, 541, 1997, 421, 727, 709, 673

Offset: 0

Jayanta Basu, Apr 18 2013

nonn

a(n) = 0 for n = 20, 165, 467, ... . Do there exist infinitely many such values of n?

These values of 0 are all conjectural. - Robert Israel, Apr 28 2021

			a(1) = 5 since 5 is the least prime that generates exactly one prime 7=5+3-1 of the given form. Again a(3) = 13 since 13 generates exactly 3 primes 17=13+5-1, 19=13+7-1 and 23=13+11-1 of the given form.

Cf. A072669, A224748.

Cn[n_] := Module[{c}, p = Prime[n]; c = 0; i = 1; While[i < n, If[PrimeQ[p + Prime[i] - 1], c = c + 1] i++]; c]; t = {};
Do[p = 0; j = 0; While[++j < 2000 && p != 1, If[Cn[j] == k, AppendTo[t, Prime[j]]; p = 1, p = 0]]; If[p == 0, AppendTo[t, 0]], {k, 0, 200}]; t

Definition clarified by Robert Israel, Apr 28 2021

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A229018 Primes of the form (3*x + 2)*2^x - 1.

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A229075 Primes of the form p^2 + q^2 + 21, where p and q are consecutive primes.

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

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A224793 Least prime p which generates exactly n primes of the form p+q-1 where q < p is prime, or 0 if (conjecturally) no such p exists.

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A229018 Primes of the form (3x + 2)2^x - 1.