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A064101 Primes p = prime(k) such that prime(k) + prime(k+5) = prime(k+1) + prime(k+4) = prime(k+2) + prime(k+3).

%I A064101 #21 Jan 09 2023 21:30:29
%S A064101 5,7,19,31,97,131,151,293,587,683,811,839,857,907,1013,1097,1279,2347,
%T A064101 2677,2833,3011,3329,4217,4219,5441,5839,5849,6113,8233,8273,8963,
%U A064101 9433,10301,10427,10859,11953,13513,13597,13721,13931,14713,15629,16057
%N A064101 Primes p = prime(k) such that prime(k) + prime(k+5) = prime(k+1) + prime(k+4) = prime(k+2) + prime(k+3).
%H A064101 Harry J. Smith, <a href="/A064101/b064101.txt">Table of n, a(n) for n = 1..1000</a>
%F A064101 Primes p = prime(k) = A000040(k) such that A359440(k+2) >= 2. - _Peter Munn_, Jan 09 2023
%e A064101 The first ten primes are 2, 3, 5, 7, 11, 13, 17, 19, 23 and 29. Take just the fourth through the ninth and rearrange them such that the first pairs with the sixth, the second with the fifth and the third with the fourth as follows: 7 and 23, 11 and 19 and 13 and 17. All three pairs sum to 30. Therefore a(2) = 7.
%p A064101 A := {}: for n to 1000 do p1 := ithprime(n); p2 := ithprime(n+1); p3 := ithprime(n+2); p4 := ithprime(n+3); p5 := ithprime(n+4); p6 := ithprime(n+5); if `and`(p1+p6 = p2+p5, p2+p5 = p3+p4) then A := `union`(A, {p1}) end if end do; A := A;
%t A064101 a = {0, 0, 0, 0, 0, 0}; Do[ a = Delete[ a, 1 ]; a = Append[ a, Prime[ n ] ]; If[ a[ [ 1 ] ] + a[ [ 6 ] ] == a[ [ 2 ] ] + a[ [ 5 ] ] == a[ [ 3 ] ] + a[ [ 4 ] ], Print[ a[ [ 1 ] ] ] ], {n, 1, 20000} ] (* RGWv *)
%t A064101 Prime[Select[Range[100], Prime[#] + Prime[# + 5] == Prime[# + 1] + Prime[# + 4] && Prime[#] + Prime[# + 5] == Prime[# + 2] + Prime[# + 3] &]]
%t A064101 Select[Partition[Prime[Range[2000]],6,1],#[[1]]+#[[6]]==#[[2]]+#[[5]] == #[[3]]+ #[[4]]&][[All,1]] (* _Harvey P. Dale_, Jan 16 2022 *)
%o A064101 (PARI) { n=0; default(primelimit, 1500000); for (k=1, 10^9, p1=prime(k) + prime(k + 5); p2=prime(k + 1) + prime(k + 4); p3=prime(k + 2) + prime(k + 3); if (p1==p2 && p2==p3, write("b064101.txt", n++, " ", prime(k)); if (n==1000, break)) ) } \\ _Harry J. Smith_, Sep 07 2009
%Y A064101 Cf. A000040, A022885, A359440.
%K A064101 easy,nonn
%O A064101 1,1
%A A064101 _Robert G. Wilson v_, Sep 17 2001