A258031 Numbers prime(k) such that D(prime(k), 3) < 0, where D( * , 3) = 3rd difference.

2, 5, 11, 19, 29, 43, 47, 59, 71, 83, 101, 109, 127, 137, 149, 151, 163, 167, 179, 191, 197, 199, 229, 239, 257, 269, 281, 283, 313, 331, 353, 367, 383, 397, 401, 419, 431, 443, 463, 487, 503, 521, 541, 557, 571, 587, 599, 601, 617, 619, 643, 647, 659, 683

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, so a(1) = prime(1) = 2;
D(prime(2), 3) = 11 - 3*7 + 3*5 - 3 > 0;
D(prime(3), 3) = 13 - 3*11 + 3*7 - 5 < 0, so a(2) = prime(3) = 5;
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, A258028, A258029, A258030.

Sequence in context: A245207 A215762 A085626 * A290164 A160536 A024979

Adjacent sequences: A258028 A258029 A258030 * A258032 A258033 A258034

Keyword

nonn,easy

Author

Clark Kimberling, Jun 05 2015

Status

approved