A258028 Numbers k such that D(prime(k), 3) > 0, where D( * , 3) = 3rd difference.

1, 3, 5, 8, 10, 14, 15, 17, 20, 23, 26, 29, 31, 33, 35, 36, 38, 39, 41, 43, 45, 46, 50, 52, 55, 57, 60, 61, 65, 67, 71, 73, 76, 78, 79, 81, 83, 86, 90, 93, 96, 98, 100, 102, 105, 107, 109, 110, 113, 114, 117, 118, 120, 124, 126, 129, 131, 134, 136, 138, 140

Offset

1,2

Comments

Partition of the positive integers: A064149, A258027, A258028;

Corresponding partition of the primes: A258029, A258030, A258031.

Links

Clark Kimberling, Table of n, a(n) for n = 1..1000

Formula

D(prime(k), 3) = P(k+3) - 3*P(k+2) + 3*P(k+1) - P(k), where P(m) = prime(m) for m >= 1.

Example

D(prime(1), 3) = 7 - 3*5 + 3*3 - 2 < 0, so a(1) = 1;
D(prime(2), 3) = 11 - 3*7 + 3*5 - 3 > 0;
D(prime(3), 3) = 13 - 3*11 + 3*7 - 5 < 0, so a(3) = 3;
D(prime(4), 3) = 17 - 3*13 + 3*11 - 7 > 0.

Mathematica

d = Differences[Table[Prime[n], {n, 1, 400}], 3];
u1 = Flatten[Position[d, 0]]  (* A064149 *)
u2 = Flatten[Position[Sign[d], 1]]   (* A258027 *)
u3 = Flatten[Position[Sign[d], -1]]  (* A258028 *)
p1 = Prime[u1] (* A258029 *)
p2 = Prime[u2] (* A258030 *)
p3 = Prime[u3] (* A258031 *)

Crossrefs

Cf. A064149, A258027, A258029, A258030, A258031.

Sequence in context: A027922 A097911 A051611 * A345427 A005004 A006218

Adjacent sequences: A258025 A258026 A258027 * A258029 A258030 A258031

Keyword

nonn,easy

Author

Clark Kimberling, Jun 05 2015

Status

approved