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Smallest members of prime 5-tuples whose sum is a prime.
1

%I #37 Sep 04 2023 16:14:10

%S 5,7,11,1481,3457,16057,19417,79687,195731,247601,266677,284737,

%T 375247,463447,465161,633791,661091,666427,829721,857951,978067,

%U 1022501,1091257,1091261,1173581,1182277,1230367,1246361,1324567,1368461,1830337,1904467,2202787,2418671

%N Smallest members of prime 5-tuples whose sum is a prime.

%C 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.

%C 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.

%C 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

%H Harvey P. Dale, <a href="/A309695/b309695.txt">Table of n, a(n) for n = 1..250</a>

%e 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.

%e 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.

%t 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 *)

%t 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 *)

%Y Cf. A022006, A022007.

%K nonn,easy

%O 1,1

%A _Philip Mizzi_, Aug 13 2019

%E More terms from _Giovanni Resta_, Aug 13 2019