A051779 -id:A051779 - cp's OEIS frontend

A125251 a(n)=sqrt(A051779(n+2)-1)/30.

5, 6, 8, 9, 14, 19, 43, 44, 77, 85, 91, 112, 113, 142, 155, 195, 196, 212, 226, 300, 308, 321, 351, 363, 399, 456, 461, 467, 485, 541, 555, 602, 604, 618, 638, 646, 720, 728, 779, 789, 891, 896, 923, 980, 1009, 1099, 1105, 1150, 1176, 1234, 1253, 1287, 1392

Offset: 1

Zak Seidov, Nov 26 2006

nonn

Consider twin primes p, q = p + 2 such that pq + 2 is prime. It would seem that there are infinitely many such p. Except for p = 3 and p = 5 all such p appear to be of the form 30k - 1 and the values of k give the current sequence. - James R. Buddenhagen, Jan 09 2007
This is true. Prime numbers (other than 2,3,5) are 30k + 1,7,11,13,17,19,23,29. p+2 is then prime only for p = 30k + 11,17,29; then p(p+2)+2 is 30k + 25,25,1 respectively, so the last case mod 30 is the only one possible. - Gareth McCaughan, Jan 09 2007
This is the sequence of positive integers k such that p = 30*k - 1, q = 30*k + 1 and p*q + 2 are all prime. - James R. Buddenhagen, Jan 09 2007

			a(1)=5 because A051779(3)=22501 and sqrt(22501-1)/30=5,
a(2)=6 because A051779(4)=32401 and sqrt(32401-1)/30=6.

Zak Seidov, Table of n, a(n) for n=1..2000

Cf. A051779.

isok(n) = isprime(p = 30*n+1) && isprime(q = 30*n-1) && isprime(p*q+2); \\ Michel Marcus, Oct 11 2013

A078597 Primes of the form p*(p+4)+2 where p and p+4 are primes.

Original entry on oeis.org

23, 79, 223, 439, 4759, 53359, 77839, 95479, 99223, 159199, 194479, 239119, 378223, 416023, 680623, 2223079, 2595319, 2873023, 3186223, 3515623, 4003999, 5022079, 6456679, 6859159, 8732023, 9235519, 9492559, 10017223, 10595023

Offset: 1

Cino Hilliard, Dec 08 2002

More generally, if a and b are even numbers, let Seq(a,b) be the sequence of primes of the form p*(p+a)+b where p and p+a are primes. Seq(a,b) is finite if either a^2+b == 2 (mod 3) or a^2-4*b is a square. Is it infinite in all other cases?

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Except for the term 23, this is a subsequence of A048880. A051779 is Seq(2, 2). A049002 is Seq(0, -2). A045637 is Seq(0, 4).

Select[ #(#+4)+2&/@Select[Prime/@Range[500], PrimeQ[ #+4]&], PrimeQ]

prodtp(n1,n2,a,b)=local(f,x); f=0; forprime(x=n1,n2,if(isprime(x+a),f=x*(x+a)+b; if(isprime(f),print(x" "x+a" "f" "); ); ); ); \ Computes that part of Seq(a,b) with n1<=p<=n2.

Edited by Dean Hickerson, Dec 10 2002

A078622 Primes of the form prime(n)prime(n2)+2.

Original entry on oeis.org

23, 67, 733, 1009, 4603, 16519, 66301, 154459, 161221, 173713, 327079, 750679, 1694809, 1940683, 2023741, 2042281, 3012169, 3852973, 4011523, 4704199, 5407561, 5536213, 7292251, 7347229, 8484901, 11359939, 11633971, 12559189

Offset: 1

Reinhard Zumkeller, Dec 11 2002

nonn

			43 = prime(14), 107 = prime(14*2) and 43*107+2 = 4603 is prime, therefore 4603 is a term.

Vincenzo Librandi, Table of n, a(n) for n = 1..5000

Cf. A051779, A048880.

[ p: n in [1..400] | IsPrime(p) where p is NthPrime(n)*NthPrime(n*2)+2 ];

Select[Table[Prime[n]Prime[2n]+2,{n,500}],PrimeQ] (* Harvey P. Dale, Nov 28 2011 *)

for(n=1,1000,p=prime(n);p2=prime(2*n);Q=p*p2+2;if(isprime(Q),print1(Q,", ")))

A108013 Primes p such that p + 2 and p*(p + 2) + 2 are primes.

Original entry on oeis.org

3, 5, 149, 179, 239, 269, 419, 569, 1289, 1319, 2309, 2549, 2729, 3359, 3389, 4259, 4649, 5849, 5879, 6359, 6779, 8999, 9239, 9629, 10529, 10889, 11969, 13679, 13829, 14009, 14549, 16229, 16649, 18059, 18119, 18539, 19139, 19379, 21599, 21839

Offset: 1

Cino Hilliard, May 30 2005

Except for the first 2 terms, these numbers all end in 9. Proof: Any odd prime p>5 can have one of the following forms: 10k+1, 10k+3, 10k+7, 10k+9.
10k+1 => p(p+2)+2 ends in 5, hence not prime, so p <> form 10k+1.
10k+3 => (p+2) ends in 5, hence not prime, so p <> form 10k+3.
10k+7 => p(p+2)+2 ends in 5, hence not prime, so p <> form 10k+7.
Thus p is of the form 10k+9 as stated. Moreover, p+2 ends in 1 and p(p+2)+2 is of the form 100h+1 since (10k+9)(10k+11)+2 = 100(k^2+2k+1)+1.
Subsequence of A051507. All terms larger than 5 are congruent to 29 mod 30. - Zak Seidov

			149*151 + 2 = 22501. 149, 151, and 22501 are all prime so 149 is in the sequence.

Michael De Vlieger, Table of n, a(n) for n = 1..5000

Cf. A051779.

[p: p in PrimesUpTo(25000)|  IsPrime(p+2) and IsPrime(p^2+2*p+2)] // Vincenzo Librandi, Jan 29 2011

Select[Prime@ Range@ 3000, AllTrue[{#2, #1 #2 + 2}, PrimeQ] & @@ {#, # + 2} &] (* Michael De Vlieger, Jan 22 2018 *)

g(n,k) = forprime(x1=3,n, x2=x1+2; if(isprime(x2), p=x1*x2+k; if(isprime(p), print1(x1",") ) ) )

A108016 Primes of the form p*(p+2)+6 where p and p+2 are primes.

Original entry on oeis.org

41, 149, 5189, 39209, 186629, 213449, 1127849, 1192469, 1695209, 2965289, 3732629, 4359749, 4460549, 5673929, 6718469, 7225349, 11370389, 12446789, 12830729, 14607689, 14837909, 16016009, 17040389, 17288969, 20684309

Offset: 1

Cino Hilliard, May 31 2005

Except for the first term, these numbers end in 9. p can take one of the forms 10k+1, 10k+3, 10k+7 or 10k+9. p = 10k+1 => p*(p+2)+6 = (10k+1)(10k+3)+6 = 10h+9. p can be 10k+1. p = 10k+3 => p+2 = 0 mod 5 not prime. p cannot be 10k+3. p = 10k+7 => p(p+2)+6 = (10k+7)(10k+9)+6 = 10h+9. p can be 10k+7. p = 10k+9 => p(p+2)+6 = (10k+9)*(10k+11)+6 = 0 mod 5 not prime. p cannot be 10k+9. Thus by exhaustion p(p+2)+6 ends in 9.

			149 = 11*13 + 6 is a term since 11, 13 and 149 are primes.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A001359, A051779.

f[p_] := p*(p + 2) + 6; f /@ Select[Range[10^4], And @@ PrimeQ[{#, # + 2, f[#]}] &] (* Amiram Eldar, Mar 26 2021 *)
Select[Times@@#+6&/@Select[Partition[Prime[Range[1000]],2,1],#[[2]]-#[[1]]==2&],PrimeQ] (* Harvey P. Dale, Aug 16 2024 *)

g(n,k=6) = forprime(x1=3,n, x2=x1+2; if(isprime(x2), p=x1*x2+k; if(isprime(p), print1(p, ", ") ) ) )

A261889 Primes that are the square of the sum of a twin prime pair plus 1.

Original entry on oeis.org

577, 1297, 7057, 14401, 41617, 90001, 147457, 156817, 484417, 746497, 1299601, 1742401, 2702737, 2944657, 4260097, 5308417, 6051601, 6780817, 8785297, 10497601, 14107537, 15210001, 16451137, 17438977, 18147601, 29419777, 38937601, 45968401, 51322897, 56791297

Offset: 1

K. D. Bajpai, Sep 05 2015

nonn

Alternatively: Primes of the form (p + q)^2 + 1 where p and q are twin primes.

All the terms are congruent to 1 (mod 3).

			577 appears in the sequence because it is a prime resulting from twin prime pair (11,13): (11 + 13)^2 + 1 = 577.
7057 appears in the sequence because it is a prime resulting from twin prime pair (41,43): (41 + 43)^2 + 1 = 7057.

K. D. Bajpai, Table of n, a(n) for n = 1..10000

Cf. A000040, A051779, A054735.

[k : p in PrimesUpTo (10000) | IsPrime(p+2) and IsPrime(k) where k is ((p + p + 2)^2 + 1)];

A261889:= proc() local a, b, d; a:= ithprime(n); b:=a+2; d:=(a+b)^2+1; if isprime(b)and isprime(d) then return (d): fi; end: seq(A261889 (), n=1..10000);

A261889 = {}; Do[p1 = Prime[n]; p2 = p1 + 2; p = (p1 + p2)^2 + 1; If[PrimeQ[p2] &&  PrimeQ[p], AppendTo[A261889, p]], {n, 1, 10000}]; A261889

forprime(p = 1,10000, if(isprime(p+2) && isprime((p + p + 2)^2 + 1), print1(( (p + p + 2)^2 + 1), ", ")));

list(lim)=my(v=List(),t,p=2); forprime(q=3,sqrtint(lim\1-1)\2+1, if(q-p==2 && isprime(t=(p+q)^2+1), listput(v,t)); p=q); Vec(v) \\ Charles R Greathouse IV, Sep 06 2015

A284045 Numbers k such that p = 6k-1 and q = 6k+1 are twin primes and (p^2 + q^2)/2 is prime.

Original entry on oeis.org

1, 25, 30, 40, 45, 70, 95, 215, 220, 385, 425, 455, 560, 565, 710, 775, 975, 980, 1060, 1130, 1500, 1540, 1605, 1755, 1815, 1995, 2280, 2305, 2335, 2425, 2705, 2775, 3010, 3020, 3090, 3190, 3230, 3600, 3640, 3895, 3945, 4455, 4480, 4615, 4900, 5045, 5495, 5525, 5750, 5880

Offset: 1

Thomas Ordowski and Altug Alkan, Mar 19 2017

nonn

Numbers k such that (6k)^2 + 1 is prime and p = 6k-1 and q = 6k+1 are twin primes.

Numbers k such that p*q + 2 is prime, where p = 6k-1 and q = 6k+1 are twin primes.

Robert G. Wilson v, Table of n, a(n) for n = 1..1000

Subsequence of A002822.

Cf. A051779.

fQ[n_] := AllTrue[{30n -1, 30n +1, 900n^2 +1}, PrimeQ]; Join[{1},5*Select[ Range@1200, fQ]] (* Robert G. Wilson v, Mar 19 2017 *)

print1(1, ", "); for(n=1, 15000, if(isprime(30*n-1)&& isprime(30*n+1)&& isprime(900*n^2+1), print1(5*n, ", ")));

a(n) == 0 (mod 5) for n > 1.

a(n+1) = 5 * A125251(n).

A108024 First instance of primes of the form p(p+2)+ k, if they exist, where p and p+2 are prime and k is an even number.

Original entry on oeis.org

17, 19, 41, 23, 47, 29, 31, 53, 163, 37, 59, 41, 43, 173, 47, 71, 53, 941, 59, 61, 83, 193, 67, 89, 71, 73, 383, 97, 79, 101, 83, 107, 89, 113, 223, 97, 227, 101, 103, 233, 107, 109, 131, 113, 137, 139, 251, 433, 127, 149, 131, 263, 137, 139, 269, 163, 167, 149, 151

Offset: 3

Cino Hilliard, May 31 2005

If p > 3 and k = 6n-2, then p(p+2) + k is composite. This follows from the fact that p and p+2 are both prime iff p = 3m+2 since p = 3m+1 => p+2 = 0 mod 3. Then p(p+2)+6n-2 = 9m^2+18m+8 + 6n-2 = 0 mod 3 composite. Therefore the above seq has no entry for k=10 = 6*2-2 because 8+10 = 0 mod 3. Similarly, if p>3, p=6m+5. As an aside, to test for twin primes > 3 we need only test numbers of the form 6m+5 = 5,11,17,23,29,..

			3*5+2 = 17,3*5+4=19,5*7+6 = 41.

Cf. A051779.

pqpk(n,m,k) = { forstep(k=2,n,2, forprime(x1=3,n, x2=x1+m; p=x1*x2+k; if(isprime(x2)&isprime(p), print1(p",");break; ) ) ) }

cp's OEIS Frontend

A125251 a(n)=sqrt(A051779(n+2)-1)/30.

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A078597 Primes of the form p*(p+4)+2 where p and p+4 are primes.

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A078622 Primes of the form prime(n)*prime(n*2)+2.

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A108013 Primes p such that p + 2 and p*(p + 2) + 2 are primes.

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A108016 Primes of the form p*(p+2)+6 where p and p+2 are primes.

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A261889 Primes that are the square of the sum of a twin prime pair plus 1.

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A284045 Numbers k such that p = 6k-1 and q = 6k+1 are twin primes and (p^2 + q^2)/2 is prime.

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A078622 Primes of the form prime(n)prime(n2)+2.