A258030 Numbers prime(k) such that D(prime(k), 3) > 0, where D( * , 3) = 3rd difference.

3, 7, 13, 23, 37, 53, 67, 73, 89, 97, 103, 107, 113, 131, 139, 157, 173, 181, 193, 211, 223, 233, 241, 263, 277, 293, 307, 311, 317, 337, 359, 373, 389, 409, 421, 433, 449, 457, 461, 479, 491, 499, 509, 523, 547, 563, 577, 593, 613, 631, 653, 661, 691, 719

Offset

1,1

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

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, A258028, A258029, A258031.

Sequence in context: A103116 A303853 A075321 * A164787 A131205 A256309

Adjacent sequences: A258027 A258028 A258029 * A258031 A258032 A258033

Keyword

nonn,easy

Author

Clark Kimberling, Jun 05 2015

Status

approved