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.

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

1,1

Comments

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.

Links

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

Example

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.

Maple

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);

Mathematica

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

Crossrefs

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

Sequence in context: A034962 A133659 A309354 * A106312 A023679 A187773

Adjacent sequences: A240722 A240723 A240724 * A240726 A240727 A240728

Keyword

nonn

Author

K. D. Bajpai, Apr 11 2014

Status

approved