A048880 -id:A048880 - cp's OEIS frontend

A051507 Primes p such that p*q+2 is prime, where q is next prime after p.

3, 5, 7, 13, 19, 67, 149, 179, 229, 239, 241, 269, 277, 307, 313, 397, 401, 419, 439, 487, 569, 613, 643, 701, 823, 863, 887, 1051, 1289, 1319, 1489, 1609, 1693, 1783, 1873, 1999, 2143, 2239, 2309, 2423, 2539, 2549, 2593, 2617, 2729, 2753, 2819

Offset: 1

N. J. A. Sloane

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A048880.

Prime[Select[Range[1000], PrimeQ[Prime[#]*Prime[# + 1] + 2] &]]
Reap[Do[If[PrimeQ[(p=Prime[k])*Prime[k+1]+2],Sow[p]],{k,1,430}]][[2,1]] (* Zak Seidov, Dec 05 2010 *)

p=2;forprime(q=3,1e4,if(isprime(p*q+2),print1(p", ")); p=q) \\ Charles R Greathouse IV, May 01 2011

A051507 = list(p for p in primes(10**4) if is_prime(p*next_prime(p)+2)) # D. S. McNeil, Dec 04 2010

A123921 Primes of form p*q - 2 where p and q are consecutive primes.

Original entry on oeis.org

13, 25589, 47051, 67589, 136889, 313589, 960389, 1005971, 1932017, 2301251, 2362331, 6100889, 6310061, 6901091, 7745051, 8236817, 9332987, 10956089, 13104389, 16850987, 19009589, 19201841, 19386371, 19998701, 20566079

Offset: 1

Ville Saalo (vsaalo(AT)iki.fi), Nov 19 2006

			211*223-2 = 47051.

T. D. Noe, Table of n, a(n) for n = 1..1000

Cf. A048797, A048880.

a123921 n = a123921_list !! (n-1)
a123921_list = filter ((== 1) . a010051) $
   map (flip (-) 2) $ zipWith (*) a000040_list (tail a000040_list)
-- Reinhard Zumkeller, Nov 11 2011

with (numtheory): for n from 1 to 1000 do if (tau(ithprime(n)*ithprime(n+1)-2)=2) then print(ithprime(n), ithprime(n+1), ithprime(n)*ithprime(n+1)-2); fi; od;

Select[Times@@@Partition[Prime[Range[1500]],2,1]-2,PrimeQ] (* Harvey P. Dale, Feb 23 2012 *)

p=2; forprime(q=3,1e5, if(isprime(t=p*q-2), print1(t", ")); p=q) \\ Charles R Greathouse IV, Apr 29 2015

A051779 Primes of form pq + 2 where p and q are twin primes.

Original entry on oeis.org

17, 37, 22501, 32401, 57601, 72901, 176401, 324901, 1664101, 1742401, 5336101, 6502501, 7452901, 11289601, 11492101, 18147601, 21622501, 34222501, 34574401, 40449601, 45968401, 81000001, 85377601, 92736901, 110880901, 118592101

Offset: 1

Joe DeMaio (jdemaio(AT)kennesaw.edu), Dec 09 1999

Starting with 3rd term, 22501, all terms are of the form 900n^2+1 with n=5, 6, 8, 9, 14, 19, 43, 44, 77, 85 (A125251). - Zak Seidov, Dec 07 2008

Primes of the form (p^2 + q^2)/2, where p and q are twin primes. - Thomas Ordowski and Altug Alkan, Mar 19 2017

			The third term 22501 is a member of the sequence because 22501=149*151+2, 22501 is prime and {149,151} is a twin prime pair.

Giovanni Resta, Table of n, a(n) for n = 1..10000

Cf. A048880, A051779, A000040, A001359, A005384, A006512, A037074, A054735.

Cf. A125251. - Zak Seidov, Dec 07 2008

with (numtheory): for n from 1 to 2000 do if (ithprime(n+1)-ithprime(n)=2) then if (tau(ithprime(n)*ithprime(n+1)+2)=2) then print(ithprime(n),ithprime(n+1), ithprime(n)*ithprime(n+1)+2); fi; fi; od;

lst={};Do[p=Prime[n];If[Length[Divisors[p-2]]==4&&(Divisors[p-2][[3]]-Divisors[p-2][[2]])==2, AppendTo[lst, p]], {n, 6*10^5}];lst (* Vladimir Joseph Stephan Orlovsky, Aug 08 2008 *)
Select[(First[#]Last[#]+2)&/@Select[Partition[Prime[Range[2700]], 2,1], Last[#]-First[#]==2&],PrimeQ]  (* Harvey P. Dale, Mar 11 2011 *)
Select[2 + Times @@@ Select[ Partition[ Prime@ Range@ 1350, 2, 1], First[#] + 2 == Last[#] &], PrimeQ] (* Robert G. Wilson v, Mar 12 2001 *)

{A037074(k) + 2} INTERSECT {A000040}. {A001359(k) * A006512(k) + 2} INTERSECT {A000040}. {A054735(k)^2 + 2*A054735(k) + 2} INTERSECT {A000040}. - Jonathan Vos Post, May 11 2006

Edited by R. J. Mathar, Aug 08 2008

A240596 Primes of the form pqr + 2 where p, q and r are consecutive primes.

Original entry on oeis.org

107, 4201, 18181981, 29884303, 72147193, 81927499, 208506511, 383148631, 402473443, 1106558899, 1391119621, 1459314919, 1498299289, 1945171369, 4593570199, 7908301729, 8052037969, 9970592521, 10594343761, 11304695329, 14119758703, 15111907009, 23157107803

Offset: 1

K. D. Bajpai, Apr 08 2014

nonn

All the terms in the sequence, except a(1), are congruent to 1 mod 6.

			107 is prime and appears in the sequence because 107 = (3*5*7)+2.
4201 is prime and appears in the sequence because 4201 = (13*17*19)+2.

K. D. Bajpai, Table of n, a(n) for n = 1..3525

Cf. A000040, A048880, A051507.

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

Select[Table[Prime[k]*Prime[k+1]*Prime[k+2]+2,{k,1,300}],PrimeQ]
Select[Times@@@Partition[Prime[Range[600]],3,1]+2,PrimeQ] (* Harvey P. Dale, Nov 21 2018 *)

s=[]; for(k=1, 1000, t=prime(k)*prime(k+1)*prime(k+2)+2; if(isprime(t), s=concat(s, t))); s \\ Colin Barker, Apr 09 2014

A342564 Numbers k such that 6k + 1 is a prime that can be written as pq + 2, with p and q being consecutive primes.

Original entry on oeis.org

6, 13, 37, 73, 793, 3750, 5400, 8893, 9600, 10082, 12150, 12973, 15913, 16537, 26533, 27335, 29400, 32413, 39853, 54150, 63037, 69337, 82835, 113437, 126142, 134085, 185852, 277350, 290400, 370513, 432553, 478837, 531037, 585937, 667333, 768980, 837013, 889350

Offset: 1

Hugo Pfoertner, Jun 20 2021

			a(1) = 6 because 6*6 + 1 = 37 can be written as 5*7 + 2.

Cf. A048880, whose first term 17 = 3*5 + 2 cannot be written as 6*k + 1.

Cf. A123921, A342565.

alist := proc(upto) local L, q, p, n, r; L := []; q := 2;
for n from 1 to upto do
    p := q; q := nextprime(p); r := p * q + 1 ;
    if modp(r, 6) = 0 and isprime(r + 1) then
       L := [op(L), iquo(r, 6)] fi od;
L end: alist(350); # Peter Luschny, Jun 20 2021

(Select[6Range[10^6]+1, PrimeQ[#] && MatchQ[FactorInteger[#-2], {{p_, 1}, {q_, 1}} /; q == NextPrime[p]]&]-1)/6 (* Jean-François Alcover, Jul 07 2021 *)

a342564(plim)={my(p1=5);forprime(p2=7,plim,my(p=p1*p2+2);if(isprime(p),print1((p-1)/6,", "));p1=p2)};
a342564(2400)

from primesieve.numpy import n_primes
from numbthy import isprime
primesarray = numpy.array(n_primes(10000,1))
for i in range (0, 9999):
    totest = int(primesarray[i] * primesarray[i+1] + 2)
    if (isprime(totest)) and  (((totest-1)%6) == 0):
        print((totest-1)//6) # Karl-Heinz Hofmann, Jun 20 2021

a(n) = (A048880(n+1) - 1)/6, excluding A048880(1).

A240715 Primes p such that pqr + 6 and pqr - 6 are primes where q and r are the next two primes after p.

Original entry on oeis.org

569, 1531, 1549, 7103, 7451, 9013, 10627, 10853, 11779, 11783, 12671, 12941, 14821, 14851, 17489, 18493, 20717, 20959, 25237, 26309, 27739, 29669, 29873, 34549, 35977, 36251, 37591, 38351, 38639, 39551, 40129, 45589, 46957, 47317, 48781, 55163, 55259

Offset: 1

K. D. Bajpai, Apr 10 2014

nonn

			569 is in the sequence because 569*571*577 + 6 = 187466729 and 569*571*577 - 6 = 187466717 are both prime where 571 and 577 are the next two primes after 569.
1531 is in the sequence because 1531*1543*1549 + 6 = 3659253823 and 1531*1543*1549 - 6 = 3659253811 are both prime where 1543 and 1549 are the next two primes after 1531.

K. D. Bajpai, Table of n, a(n) for n = 1..1444

Cf. A006094, A048880, A051507, A240596.

[p: p in PrimesUpTo(10^5) | IsPrime(t-6) and IsPrime(t+6) where t is p*NextPrime(p)*NextPrime(NextPrime(p))]; // Bruno Berselli, Apr 11 2014

KD := proc(n) local a,b,d; a:=ithprime(n)*ithprime(n+1)*ithprime(n+2); b:=a+6; d:=a-6; if  isprime(b) and isprime(d) then RETURN (ithprime(n)); fi; end: seq(KD(n), n=1..10000);

c = 0; Do[If[PrimeQ[Prime[n]*Prime[n+1]*Prime[n+2] +6] && PrimeQ[Prime[n]*Prime[n+1]*Prime[n+2] -6],c=c+1;Print[c, " ", Prime[n]]],{n,1,500000}];
KD={};   f=Prime[n+1]*Prime[n+2];  Do[p=Prime[n]; If[ PrimeQ[p*f+6] && PrimeQ[p*f-6], AppendTo[KD,p]], {n,10000}]; KD
Select[Partition[Prime[Range[6000]],3,1],AllTrue[Times@@#+{6,-6},PrimeQ]&][[All,1]] (* Harvey P. Dale, Oct 29 2022 *)

A342565 Numbers k such that 6k - 1 is a prime that can be written as pq - 2, with p and q being consecutive primes.

Original entry on oeis.org

4265, 7842, 11265, 22815, 52265, 160065, 167662, 322003, 383542, 393722, 1016815, 1051677, 1150182, 1290842, 1372803, 1555498, 1826015, 2184065, 2808498, 3168265, 3200307, 3231062, 3333117, 3427680, 3676962, 3913915, 4042598, 4323102, 4537907, 4623542, 4798955

Offset: 1

Hugo Pfoertner, Jun 20 2021

			a(1) = 4265, because the prime 25589 = 6*4265 - 1 can be written as 157*163 - 2, with 157 and 163 being consecutive primes.

Cf. A123921, whose first term 13 = 3*5 - 2 cannot be written as 6*k - 1.

Cf. A048880, A342564.

a342565(plim)={my(p1=5);forprime(p2=7,plim,my(p=p1*p2-2);if(isprime(p),print1((p+1)/6,", "));p1=p2)};
a342565(5400)

from primesieve.numpy import n_primes
from numbthy import isprime
primesarray = numpy.array(n_primes(10000, 1))
for i in range (0, 9999):
    totest = int(primesarray[i] * primesarray[i+1] - 2)
    if (isprime(totest)) and  (((totest+1)%6) == 0):
        print((totest+1)//6) # Karl-Heinz Hofmann, Jun 20 2021

a(n) = (A123921(n+1) + 1)/6, excluding A123921(1).

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,", ")))

A229570 Primes of form p*q + 30, where p and q are consecutive primes.

Original entry on oeis.org

107, 173, 251, 353, 467, 929, 2521, 4787, 7417, 8663, 10433, 12347, 17977, 19073, 25621, 28921, 32429, 39233, 42019, 50651, 55717, 60521, 77867, 95507, 97373, 99251, 111577, 116969, 126757, 131783, 141397, 159227, 164039, 171401, 186653, 194507, 198937, 205223

Offset: 1

K. D. Bajpai, Sep 26 2013

nonn

Conjecture: The expression p*q + c with p and q consecutive primes and c = 30 generates more primes than any other value of c in the range 1 < c < 100 and p = 48611 which is 5000th prime. Hence, c = 30 is considered for this sequence.

			a(1)=107: prime(4)*prime(5)+30=107, which is prime.
a(6)=929: prime(10)*prime(11)+30=929, which is prime.

K. D. Bajpai, Table of n, a(n) for n = 1..1000

Cf. A048880.

KD:= proc() local a; a:= ithprime(n)*ithprime(n+1)+30; if isprime((a)) then RETURN((a)):fi;end: seq(KD(),n=1..500);

Select[Table[Prime[n]*Prime[n+1]+30,{n,100}],PrimeQ]

cp's OEIS Frontend

A051507 Primes p such that p*q+2 is prime, where q is next prime after p.

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A123921 Primes of form p*q - 2 where p and q are consecutive primes.

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A051779 Primes of form pq + 2 where p and q are twin primes.

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A240596 Primes of the form p*q*r + 2 where p, q and r are consecutive primes.

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A342564 Numbers k such that 6*k + 1 is a prime that can be written as p*q + 2, with p and q being consecutive primes.

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A240715 Primes p such that p*q*r + 6 and p*q*r - 6 are primes where q and r are the next two primes after p.

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Magma

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A342565 Numbers k such that 6*k - 1 is a prime that can be written as p*q - 2, with p and q being consecutive primes.

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Python

A240596 Primes of the form pqr + 2 where p, q and r are consecutive primes.

A342564 Numbers k such that 6k + 1 is a prime that can be written as pq + 2, with p and q being consecutive primes.

A240715 Primes p such that pqr + 6 and pqr - 6 are primes where q and r are the next two primes after p.

A342565 Numbers k such that 6k - 1 is a prime that can be written as pq - 2, with p and q being consecutive primes.

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