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%I #21 Jan 09 2023 21:30:29
%S 5,7,19,31,97,131,151,293,587,683,811,839,857,907,1013,1097,1279,2347,
%T 2677,2833,3011,3329,4217,4219,5441,5839,5849,6113,8233,8273,8963,
%U 9433,10301,10427,10859,11953,13513,13597,13721,13931,14713,15629,16057
%N Primes p = prime(k) such that prime(k) + prime(k+5) = prime(k+1) + prime(k+4) = prime(k+2) + prime(k+3).
%H Harry J. Smith, <a href="/A064101/b064101.txt">Table of n, a(n) for n = 1..1000</a>
%F Primes p = prime(k) = A000040(k) such that A359440(k+2) >= 2. - _Peter Munn_, Jan 09 2023
%e 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 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 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 Prime[Select[Range[100], Prime[#] + Prime[# + 5] == Prime[# + 1] + Prime[# + 4] && Prime[#] + Prime[# + 5] == Prime[# + 2] + Prime[# + 3] &]]
%t Select[Partition[Prime[Range[2000]],6,1],#[[1]]+#[[6]]==#[[2]]+#[[5]] == #[[3]]+ #[[4]]&][[All,1]] (* _Harvey P. Dale_, Jan 16 2022 *)
%o (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 Cf. A000040, A022885, A359440.
%K easy,nonn
%O 1,1
%A _Robert G. Wilson v_, Sep 17 2001
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