login
A258031
Numbers prime(k) such that D(prime(k), 3) < 0, where D( * , 3) = 3rd difference.
5
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
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
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, Jun 05 2015
STATUS
approved