A051507 -id:A051507 - cp's OEIS frontend

A048880 Primes of form pq+2 where p and q are consecutive primes.

17, 37, 79, 223, 439, 4759, 22501, 32401, 53359, 57601, 60493, 72901, 77839, 95479, 99223, 159199, 164011, 176401, 194479, 239119, 324901, 378223, 416023, 497011, 680623, 756853, 804511, 1115113, 1664101, 1742401, 2223079

Offset: 1

Herman H. Rosenfeld (herm3(AT)pacbell.net)

All terms > 17 are congruent to 1 mod 6. - Zak Seidov, Dec 03 2010

			487*491+2=239119.

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

Cf. A051507, A123921, A342564.

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;

Reap[Do[If[PrimeQ[p=Prime[k]*Prime[k+1]+2],Sow[p]],{k,1,430}]][[2,1]] (* Zak Seidov Dec 03 2010 *)

a(n) = 6*A342564(n-1) + 1 for n >= 2. - Hugo Pfoertner, Jun 24 2021

Corrected and extended by Joe DeMaio (jdemaio(AT)kennesaw.edu).

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

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 *)

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",") ) ) )

A224789 Primes p such that both p + nextprime(p) + 1 and p*nextprime(p) + 2 are primes.

Original entry on oeis.org

5, 7, 13, 19, 67, 229, 269, 307, 313, 401, 439, 613, 643, 863, 1051, 1693, 1783, 1999, 2143, 2239, 2309, 2423, 2549, 2753, 2819, 3037, 3079, 3089, 3361, 3373, 3389, 3677, 3863, 3877, 4139, 4259, 4519, 4663, 4909, 4933, 5323, 5527, 5581, 5849, 6359, 6577

Offset: 1

Jayanta Basu, Apr 17 2013

nonn

Intersection of A051507 and A177017.

			5 is a member since 5 + 7 + 1 = 13 and 5 * 7 + 2 = 37 are both primes.

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

Cf. A051507, A177017.

Select[Prime[Range[900]], PrimeQ[# + NextPrime[#] + 1] && PrimeQ[#*NextPrime[#] + 2] &]
npQ[n_]:=Module[{np=NextPrime[n]},AllTrue[{n+np+1,n*np+2},PrimeQ]]; Select[ Prime[ Range[900]],npQ] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Jul 04 2017 *)

A240618 Primes p such that pqr + 2 is prime, where q and r are the next two primes after p.

Original entry on oeis.org

3, 13, 257, 307, 409, 431, 587, 719, 733, 1031, 1109, 1123, 1129, 1237, 1657, 1987, 1999, 2143, 2179, 2239, 2411, 2467, 2843, 3041, 3191, 3433, 3691, 3943, 4051, 4219, 4289, 4561, 4567, 4721, 4817, 4831, 4943, 4993, 5521, 5563, 5623, 5689, 5813, 6257, 6983, 7043

Offset: 1

K. D. Bajpai, Apr 09 2014

nonn

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

K. D. Bajpai, Table of n, a(n) for n = 1..6606

Cf. A000040, A048880, A051507, A240596.

KD := proc(n) local a,b; a:=ithprime(n)*ithprime(n+1)*ithprime(n+2); b:=a+2; if  isprime(b) then RETURN (ithprime(n)); fi; end: seq(KD(n), n=1..2000);

KD={}; Do[p=Prime[n]; If[PrimeQ[p*Prime[n+1]*Prime[n+2] + 2], AppendTo[KD,p]], {n,1000}]; KD

A240621 Primes p such that pq + 6 and pq - 6 are primes where p and q are consecutive primes.

Original entry on oeis.org

5, 7, 11, 19, 23, 37, 79, 97, 307, 349, 439, 479, 503, 719, 907, 983, 991, 1061, 1069, 1109, 1597, 1609, 1621, 1867, 2111, 2609, 3301, 3371, 3851, 4129, 4211, 4639, 4999, 5119, 5471, 5683, 5779, 5867, 5939, 6563, 7951, 9337, 9461, 9551, 10061, 10181, 10273, 12251

Offset: 1

K. D. Bajpai, Apr 09 2014

nonn

			7 is in the sequence because 7*11 + 6 = 83 and 7*11 - 6 = 71 are both prime where 7 and 11 are consecutive primes.
37 is in the sequence because 37*41 + 6 = 1523 and 37*41 - 6 = 1511 are both prime where 37 and 41 are consecutive primes.

K. D. Bajpai, Table of n, a(n) for n = 1..1331

Cf. A006094, A048880, A051507, A240596.

KD := proc(n) local a,b,d; a:=ithprime(n)*ithprime(n+1); b:=a+6; d:=a-6; if  isprime(b) and isprime(d) then RETURN (ithprime(n)); fi; end: seq(KD(n), n=1..5000);

Select[Partition[Prime[Range[1500]],2,1],AllTrue[Times@@#+{6,-6},PrimeQ]&][[All,1]] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Jun 21 2017 *)

isok(p) = isprime(p) && (q = nextprime(p+1)) && isprime(p*q-6) && isprime(p*q+6); \\ Michel Marcus, Apr 12 2014

A240725 Primes p such that p^2q^2r^2 + 12 and p^2q^2r^2 - 12 are primes where q and r are next two primes after p.

Original entry on oeis.org

23, 31, 43, 521, 1061, 2153, 3457, 4019, 4943, 5477, 6991, 7577, 8291, 8539, 10993, 11953, 14767, 17957, 18439, 26321, 40993, 41047, 53269, 57917, 71347, 79979, 80989, 88997, 91499, 92269, 94561, 108457, 109111, 112019, 117671, 121763, 133103, 140407, 147073

Offset: 1

K. D. Bajpai, Apr 11 2014

nonn

In the expression prime(n)^2 * prime(n+1)^2 * prime(n+2)^2 +/- c, c = 12 is the smallest integer that yields a sequence of such primes. That means for c = 1...11 no such sequence with a large number of primes is obtained.

			23 is prime and appears in the sequence because 23^2 * 29^2 * 31^2 + 12 = 427538341 and 23^2 * 29^2 * 31^2 - 12 = 427538317 are both prime where 29 and 31 are the next two primes after 23.
31 is prime and appears in the sequence because 31^2 * 37^2 * 41^2 + 12 = 2211538741 and 31^2 * 37^2 * 41^2 - 12 = 2211538717 are both prime where 37 and 41 are the next two primes after 31.

K. D. Bajpai, Table of n, a(n) for n = 1..1383

Cf. A006094, A048880, A051507, A240596, A240715.

KD := proc(n) local a, b, d; a:=ithprime(n)^2*ithprime(n+1)^2*ithprime(n+2)^2; b:=a+12; d:=a-12; if  isprime(b) and isprime(d) then RETURN (ithprime(n)); fi; end: seq(KD(n), n=1..20000);

c=0;Do[If[PrimeQ[Prime[n]^2*Prime[n+1]^2*Prime[n+2]^2+12]&&PrimeQ[Prime[n]^2*Prime[n+1]^2*Prime[n+2]^2-12],c=c+1;Print[c," ", Prime[n]]], {n,1,1000000}];
KD = {}; Do[p = Prime[n]; If[PrimeQ[Prime[n]^2*Prime[n + 1]^2*Prime[n + 2]^2 + 12] && PrimeQ[Prime[n]^2*Prime[n + 1]^2*Prime[n + 2]^2 - 12], AppendTo[KD, p]], {n, 10000}]; KD

A340468 a(n) is the least prime of the form 2 + Product_{i=n..m} prime(i).

Original entry on oeis.org

5, 7, 79, 13, 223, 19, 439, 130753887906569681111538991218568790437537693430279000532630035672131604633987039552816424896353327834998483765849409837393409377729040653460715050958787058270805333463, 31, 34826927179023475480751694965449235272424989980919

Offset: 2

Robert Israel, Jan 08 2021

nonn

If n is in A029707, a(n) = 2+prime(n).
If n is not in A029707 but prime(n) is in A051507, a(n) = 2+prime(n)*prime(n+1).
a(15) > 10^1000 if it exists.

			a(2) = 2+3 = 5.
a(3) = 2+5 = 7.
a(4) = 2+7*11 = 79.
a(5) = 2+11 = 13.
a(6) = 2+13*17 = 223.
a(7) = 2+17 = 19.
a(8) = 2+19*23 = 439.
a(9) = 2+23*29*...*431.

Robert Israel, Table of n, a(n) for n = 2..14

Cf. A029707, A051507.

f:= proc(n) local i,t;
  t:= 1;
  for i from n do
    t:= t*ithprime(i);
    if isprime(t+2) then return t+2 fi;
  od
end proc:
seq(f(n),n=2..14);

from sympy import isprime, nextprime, prime
def a(n):
  prodpnpm = pm = prime(n)
  while not isprime(2+prodpnpm): pm = nextprime(pm); prodpnpm *= pm
  return 2+prodpnpm
print([a(n) for n in range(2, 12)]) # Michael S. Branicky, Jan 08 2021

cp's OEIS Frontend

A048880 Primes of form pq+2 where p and q are consecutive 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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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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A108013 Primes p such that p + 2 and p*(p + 2) + 2 are primes.

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A224789 Primes p such that both p + nextprime(p) + 1 and p*nextprime(p) + 2 are primes.

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A240618 Primes p such that p*q*r + 2 is prime, where q and r are the next two primes after p.

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A240621 Primes p such that p*q + 6 and p*q - 6 are primes where p and q are consecutive primes.

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

A240618 Primes p such that pqr + 2 is prime, where q and r are the next two primes after p.

A240621 Primes p such that pq + 6 and pq - 6 are primes where p and q are consecutive primes.

A240725 Primes p such that p^2q^2r^2 + 12 and p^2q^2r^2 - 12 are primes where q and r are next two primes after p.