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

5, 7, 19, 31, 97, 131, 151, 293, 587, 683, 811, 839, 857, 907, 1013, 1097, 1279, 2347, 2677, 2833, 3011, 3329, 4217, 4219, 5441, 5839, 5849, 6113, 8233, 8273, 8963, 9433, 10301, 10427, 10859, 11953, 13513, 13597, 13721, 13931, 14713, 15629, 16057

Offset: 1

Author

Robert G. Wilson v, Sep 17 2001

Keywords

• easy
• nonn

Examples

			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.
		

Links

• Harry J. Smith, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A000040, A022885, A359440.

Programs

Maple

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;

Mathematica

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 *)
Prime[Select[Range[100], Prime[#] + Prime[# + 5] == Prime[# + 1] + Prime[# + 4] && Prime[#] + Prime[# + 5] == Prime[# + 2] + Prime[# + 3] &]]
Select[Partition[Prime[Range[2000]],6,1],#[[1]]+#[[6]]==#[[2]]+#[[5]] == #[[3]]+ #[[4]]&][[All,1]] (* Harvey P. Dale, Jan 16 2022 *)

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

Formula

Primes p = prime(k) = A000040(k) such that A359440(k+2) >= 2. - Peter Munn, Jan 09 2023