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%I #5 Jun 05 2015 12:59:20
%S 2,4,6,9,12,16,19,21,24,25,27,28,30,32,34,37,40,42,44,47,48,51,53,56,
%T 59,62,63,64,66,68,72,74,77,80,82,84,87,88,89,92,94,95,97,99,101,103,
%U 106,108,112,115,119,121,125,128,130,133,135,137,139,141,143
%N Numbers k such that D(prime(k), 3) < 0, where D( * , 3) = 3rd difference.
%C Partition of the positive integers: A064149, A258027, A258028;
%C Corresponding partition of the primes: A258029, A258030, A258031.
%H Clark Kimberling, <a href="/A258027/b258027.txt">Table of n, a(n) for n = 1..1000</a>
%F 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.
%e D(prime(1), 3) = 7 - 3*5 + 3*3 - 2 < 0;
%e D(prime(2), 3) = 11 - 3*7 + 3*5 - 3 > 0, so a(1) = 2;
%e D(prime(3), 3) = 13 - 3*11 + 3*7 - 5 < 0;
%e D(prime(4), 3) = 17 - 3*13 + 3*11 - 7 > 0, so a(2) = 4.
%t d = Differences[Table[Prime[n], {n, 1, 400}], 3];
%t u1 = Flatten[Position[d, 0]] (* A064149 *)
%t u2 = Flatten[Position[Sign[d], 1]] (* A258027 *)
%t u3 = Flatten[Position[Sign[d], -1]] (* A258028 *)
%t p1 = Prime[u1] (* A258029 *)
%t p2 = Prime[u2] (* A258030 *)
%t p3 = Prime[u3] (* A258031 *)
%Y Cf. A064149, A258028, A258029, A258030, A258031.
%K nonn,easy
%O 1,1
%A _Clark Kimberling_, Jun 05 2015
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