OFFSET
1,1
COMMENTS
A prime 5-tuple is a set of five prime numbers of one of two forms: (p, p+2, p+6, p+8, p+12) and (p, p+4, p+6, p+10, p+12), with sums 5p+28 and 5p+32, respectively.
For either form, the sum of the five members sometimes yields a prime. It has been observed that a prime sum is yielded less frequently by quintuplets of the first form than by those of the second.
In the first 250 terms, quintuplets of the first form (see the preceding comments) represent just over 46% of the terms and quintuplets of the second form represent just over 53% of the terms. - Harvey P. Dale, Sep 04 2023
LINKS
Harvey P. Dale, Table of n, a(n) for n = 1..250
EXAMPLE
For p=5, (p, p+2, p+6, p+8, p+12) = (5,7,11,13,17): all primes, hence a prime 5-tuple. Their sum, 5p+28 = 53, is a prime, so 5, the smallest member, is a term of the sequence.
For p=7, (p, p+2, p+6, p+8, p+12) = (7,9,13,15,19); not all members are prime, but (p, p+4, p+6, p+10, p+12) = (7,11,13,17,19), all primes, hence a prime 5-tuple; their sum, 5p+32 = 67, is a prime, so 7, the smallest member, is a term of the sequence.
MATHEMATICA
qp[p_] := AllTrue[p + {2, 6, 8, 12, 28 + 4 p}, PrimeQ] || AllTrue[p + {4, 6, 10, 12, 32 + 4 p}, PrimeQ]; Select[Prime@ Range[180000], qp] (* Giovanni Resta, Aug 13 2019 *)
Select[Partition[Prime[Range[180000]], 5, 1], PrimeQ[Total[#]]&&(Differences[#]=={2, 4, 2, 4} || Differences[#]=={4, 2, 4, 2})&][[;; , 1]] (* Harvey P. Dale, Sep 04 2023 *)
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Philip Mizzi, Aug 13 2019
EXTENSIONS
More terms from Giovanni Resta, Aug 13 2019
STATUS
approved