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A258030
Numbers prime(k) such that D(prime(k), 3) > 0, where D( * , 3) = 3rd difference.
5
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
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
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, Jun 05 2015
STATUS
approved