A048880 -id:A048880 - cp's OEIS frontend

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

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

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

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

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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.

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cp's OEIS Frontend

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

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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.